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Complete sets of disjoint difference families and their applications
Journal of Statistical Planning and Inference 106 (2002) 87 – 103
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Complete sets of disjoint di%erence families and their applications R. Fuji-Haraa; ∗ , Y. Miaoa , S. Shinoharab a Institute
b Doctoral
of Policy and Planning Sciences, University of Tsukuba, Tsukuba 305-8573, Japan Program in Policy and Planning Sciences, University of Tsukuba, Tsukuba 305-8573, Japan Received 2 December 1998; accepted 15 February 1999 Dedicated to the memory of Professor Sumiyasu Yamamoto
Abstract Let G be an abelian group. A collection of (G; k; ) disjoint di%erence families, {F0 ; F1 ; : : : ; Fs−1 }, is a complete set of disjoint di%erence families if 06i6s−1 B∈Fi B form a partition of G − {0}. In this paper, several construction methods are provided for complete sets of disjoint di%erence families. Applications to one-factorizations of complete graphs and to cyclic 2002 Elsevier Science B.V. cally resolvable cyclic Steiner triple systems are also described. All rights reserved. Keywords: Complete set of disjoint families; Cyclically resolvable cyclic Steiner system; One-factorization of a complete graph
1. Introduction Let G be an abelian group of order v; k an integer satisfying 2 6 k 6 v, and a positive integer. A (G; k; ) di&erence family, denoted by (G; k; )-DF, is a collection F = {Bi : i ∈ I } of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely ways as a di%erence of two elements lying in some base blocks of F. The number of base blocks of a (G; k; )-DF is obviously (|G|−1)=(k(k −1)), and hence a necessary condition for the existence of a (G; k; )-DF is that (|G| − 1) ≡ 0 (mod k(k − 1)) holds. If the base blocks of a (G; k; )-DF are mutually disjoint, then this (G; k; )-DF is said to be disjoint, and denoted by (G; k; )-DDF. (G; k; )-DFs and (G; k; )-DDFs have been investigated intensively, see ∗
Corresponding author. E-mail addresses: [email protected] (R. Fuji-Hara), [email protected] (Y. Miao), [email protected] (S. Shinohara). c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 2 0 5 - 7
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for example (Abel, 1996; Dinitz and Rodney, 1997; Dinitz and Shalaby, to appear; Furino et al., 1996; Wilson, 1972). Let {F0; F1 ; : : : ; Fs−1 } be a collection of (G; k; ) disjoint di%erence families. If 06i6s−1 B∈Fi B form a partition of G − {0}, then the collection of (G; k; )-DDFs {F0 ; F1 ; : : : ; Fs−1 } is called a complete set of disjoint di%erence families and denoted by (G; k; )-CDDF, whereeach Fi ; 0 6 i 6 s − 1, is the component of the (G; k; )-CDDF. Clearly, {B: B ∈ 06i6s−1 Fi } forms a (G; k; s)-DDF, while the number s of component disjoint di%erence families therein is (k − 1)=. When = k − 1, a (G; k; )-CDDF is just a (G; k; )-DDF. Complete sets of disjoint di%erence families are of interest in their own right, as well as having applications in the construction of other types of combinatorial structures. Starters in an abelian group G are in fact (G; 2; 1)-CDDFs, which have been used to construct Room squares, round robin tournaments, one-rotational one-factorizations of complete graphs, etc., see for example Dinitz (1996). (G; k; k − 1)-CDDFs are just (G; k; k − 1)-DDFs, which have been used to construct resolvable balanced incomplete block designs, see for example Furino et al. (1996). Complete sets of disjoint di%erence families also have new applications which will be described later. In this paper, we present several constructions for complete sets of disjoint di%erence families, mainly with = k − 1. Composition theorems are used to produce new complete sets of disjoint di%erence families from two collections of “small” ones by means of di%erence matrices. A computer is used to construct two individual examples of (Z6t+1 ; 3; 1)-CDDFs, where t = 4 and 9. Some complete sets of disjoint di%erence families over Hnite Helds and Hnite rings are constructed directly by exploiting some properties of Hnite Helds and Hnite rings. Two applications of complete sets of disjoint di%erence families are also mentioned: a (G; 2; 1)-CDDF is used to construct a one-factorization of the complete graph based on Z2 ⊕ G which is invariant under the automorphism group Z2 ⊕ G, while a (G; 3; 1)-CDDF is used to construct a cyclically resolvable cyclic Steiner triple system of order 3|G|. 2. Composition theorems Before describing any method of construction, we Hrst make a simple observation on complete sets of disjoint di%erence families. Let G be an abelian group. If there is a (G; k; )-CDDF, where is a factor of k −1, then by combining its n (1 6 n 6 s) component (G; k; )-DDFs into one (G; k; n)-DDF, we can get a (G; k; n)-CDDF, provided that n is still a factor of k − 1. Jungnickel (1978), Jimbo (1993), and Kageyama and Miao (1998) presented several composition theorems for di%erence families with or without other properties. The following are their variations for complete sets of disjoint di%erence families. Here a (G; k; ) di&erence matrix, denoted by (G; k; )-DM, is a k × |G| matrix (mij ) with entries from an abelian group G such that the list of di%erences (mis −mjs : 0 6 s 6 |G|−1) contains each element of G precisely times, where i = j and 0 6 i; j 6 k −1. Di%erence matrices have been studied widely, see for example, (Colbourn and de Launey (1996)). The following simple assertion can be found in Colbourn and Colbourn (1984).
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Lemma 2.1. (Colbourn and Colbourn; 1984). Let ij ≡ ij (mod w) for i=0; 1; : : : ; k −1; and j = 0; 1; : : : ; w − 1; where gcd((k − 1)!; w) = 1. Then = (ij ) is a (Zw ; k; 1)-DM. In particular; if w is a prime number; then there exists a (Zw ; k; 1)-DM for any integer k (6 w). Di%erence matrices are useful in the construction of complete sets of disjoint di%erence families. Theorem 2.2. Let G1 and G2 be two abelian groups. If there exist a (G1 ; k; )-CDDF; a (G2 ; k; )-CDDF and a (G2 ; k + 1; 1)-DM; then there exists a (G1 ⊕ G2 ; k; )-CDDF. Proof. First of all; we note that the property of the di%erence matrix is conserved even if we add an element to any columns or any rows. Thus; without loss of generality; we may assume that in a di%erence matrix; the elements in the Hrst row and in the Hrst column are all 0’s. l Let {D0 ; D1 ; : : : ; Ds−1 } be the (G1 ; k; )-CDDF with Dl = {D0l ; D1l ; : : : ; Dr−1 }; Dil = {dli0 ; : : : ; dli; k−1 }; 0 6 l 6 s − 1; 0 6 i 6 r − 1, where r = (|G1 | − 1)=(k(k − 1)); l }; s=(k −1)=. Let {E0 ; E1 ; : : : ; Es−1 } be the (G2 ; k; )-CDDF with El ={E0l ; E1l ; : : : ; Et−1 l l l Ej = {ej0 ; : : : ; ej; k−1 }; 0 6 l 6 s − 1; 0 6 j 6 t − 1, where t = (|G2 | − 1)=(k(k − 1)). Further, let A = (aij ); 0 6 i 6 k − 1; 0 6 j 6 |G2 | − 1 be the (G2 ; k; 1)-DM derived from the (G2 ; k + 1; 1)-DM by removing its Hrst row. Put Fl = {Fijl : 0 6 i 6 r − 1; 0 6 j 6 |G2 | − 1} with Fijl = {(dli0 ; a0j ); : : : ; (dli; k−1 ; ak−1; j )};
El = {Eil : 0 6 i 6 t − 1}
l with Eil = {(0; ei0 ); : : : ; (0; ei;l k−1 )}
and let Fl be the union of Fl and El . Kageyama and Miao (1998) proved that Fl is a (G1 ⊕ G2 ; k; )-DDF. So we only need to prove that all the base blocks of Fl ; l = 0; : : : ; s − 1, form a partition of G1 ⊕ G2 − {(0; 0)}. We are to prove that each element is distinct from others. If (dli m ; am j ) = (dlim ; amj ) then dli m = dlim and am j = amj . By the deHnition of a complete set of disjoint di%erence families, we can conclude that l = l; i = i and m = m. Then, by the property of the di%erence matrix A, we should have j = j . The cases of (dlim ; amj ) = (0; eil m ) are impossible since dlim = 0. l l If (0; eim = eil m . By the deHnition of a complete set of disjoint ) = (0; eil m ), then eim di%erence families, l = l ; i = i and m = m should also hold. Since these elements are all not (0; 0), and are just all the elements of Fl ; l = 0; : : : ; s − 1, they form a partition of G1 ⊕ G2 − {(0; 0)}. The construction method of Theorem 2.2 does not contain the construction of complete sets of cyclic disjoint di%erence families (i.e., the abelian group G is cyclic). However, the structures of groups are important to our later applications. So we also need the following construction.
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Theorem 2.3. Let G2 be a subgroup of an abelian group G such that G=G2 is isomorphic to an abelian group G1 of order being not k. If there exist a (G1 ; k; )CDDF; a (G2 ; k; )-CDDF and a (G2 ; k + 1; 1)-DM; then there exists a (G; k; )CDDF. This theorem can be proved in a similar way to that of Theorem 2.2, while the construction of corresponding (G; k; )-DDF can be found in Theorem 6.2 of Kageyama and Miao (1998). We also note that Theorem 2.3 can be extended to the case of a non-abelian group. Of course, the notions of a complete set of disjoint di%erence families and a di%erence matrix should be extended simultaneously to the case of a non-abelian group. We will not go further here. 3. Direct constructions In order for the composition theorems described in Section 2 to work, we need several complete sets of “small” disjoint di%erence families to start with. We will describe direct constructions for such kind of di%erence families in this section, most of which are based on Hnite Helds. We Hrst consider those cases where no prime Hnite Held can exist. In such cases, the commonly used technique is to use a computer to search their solutions. We succeeded in Hnding the smallest two cases for k = 3, i.e., a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF. Note that although a Hnite Held of order 25 exists, and 25 ≡ 1 (mod 6), a prime Hnite Held of order 25 does not exist. Lemma 3.1. There exist a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF. Proof. The following two collections of blocks are base blocks of a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF; respectively: {{{5; 12; 10}; {21; 13; 9}; {18; 17; 7}; {3; 6; 22}} ; {{4; 15; 2}; {19; 24; 23}; {1; 16; 8}; {20; 11; 14}}} and {{{6; 27; 47}; {31; 25; 50}; {14; 12; 51}; {53; 45; 48}; {33; 16; 7}; {8; 20; 19}; {54; 30; 26}; {42; 9; 2}; {28; 5; 15}} ; {{34; 36; 21}; {41; 46; 38}; {22; 52; 29}; {35; 4; 24}; {18; 1; 17}; {49; 13; 39}; {3; 44; 40}; {23; 11; 32}; {37; 10; 43}}} :
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We have used the terminology “list” in the deHnition of a di%erence matrix. By a list we mean a collection of elements in which each element occurs nonnegative times. Note that “list” is also called “multiset” sometimes. We use the notation (x1 ; : : : ; x‘ ). The orderis not taken into account in our lists. If Xi ; i = 1; 2; : : : ; t, are lists, then the t notation i=1 Xi is used to denote the concatenation of the lists. In some case it can be determined whether or not an arbitrary collection of blocks F will be a di%erence family, by the following procedure. Let B be a subset of an abelian group G. Then deHne the list of di&erence from B to be the list MB = (a − b: a; b ∈ B; a = b). When F = {Bi : i ∈ I } is a family of k-subsets of G, we deHne MF = i∈I MBi . If MF contains every nonzero element of G exactly times, then F is a (G; k; )-DF. Note that we here only consider those di%erence families without short blocks. We Hrst consider the constructions of complete sets of disjoint di%erence families in G(q), the additive group of GF(q). For convenience, we select and Hx, for each prime power q, a primitive element ! of GF(q). When e|(q − 1), we deHne the cosets e modulo the eth power, H0e = H e ; H1e ; : : : ; He−1 , by Hme = {!t : t ≡ m (mod e)}. We read e e : H e is the multhe subscripts modulo e, so that if a ∈ Hm and b ∈ Hne , then a · b ∈ Hm+n e tiplicative subgroup of GF(q) − {0} of order (q − 1)=e and index e. {H e ; H1e ; : : : ; He−1 } are the cyclotomic classes of GF(q) of index e. They evidently partition GF(q) − {0}. e The cyclotomic classes {H e ; H1e ; : : : ; He−1 } will be denoted by He . Given a set of e distinct elements in GF(q), if they belong to e di%erent cyclotomic e classes H e ; H1e ; : : : ; He−1 , then we say that this set of e elements forms a system of e }, and is denoted distinct representatives for the cyclotomic classes {H e ; H1e ; : : : ; He−1 e by SDRC(H ). Note that if q is even, then −1 = 1 is always an eth power in GF(q). If q is odd, then −1 ∈ H e if and only if 2e|(q − 1). In fact, −1 = !(q−1)=2 is an eth power if and only if (q − 1)=2 ≡ 0 (mod e). It will be convenient to introduce a multiplication of lists as follows: (ai : i ∈ I ) · (bj : j ∈ J ) = (ai · bj : i ∈ I; j ∈ J ):
Lemma 3.2. If there exists a (G(q); k; )-CDDF; then there exists a (G(q n ); k; )CDDF for every n ∈ N. Proof. Let {F0 ; : : : ; Fs−1 } be a (G(q); k; )-CDDF; where F‘ = {Bi‘ : i ∈ I (‘)}; 06 ‘ 6 s − 1; s = (k − 1)=; are the component (G(q); k; )-DDFs; such that MF‘ = ‘ ‘ ‘ i∈I (‘) MBi = (GF(q) − {0}) and Bi ∩ Bj = ∅ for all i; j ∈ I (l) where i = j; and for all ‘; 0 6 ‘ 6 s − 1; and also 06l6s−1 i∈I (l) Bil = GF(q) − {0}. Consider GF(q) as a subHeld of GF(q n ); then GF(q) − {0} is the subgroup H e of the eth powers in GF(q n ) where e = (q n − 1)=(q − 1). Now let R be any system of distinct representatives for the cosets He modulo H e in GF(q n ). Then R is a set of Held elements such that R·H e =GF(q n )−{0}. Consider the families F‘∗ ={rBi‘ : i ∈ I (‘); r ∈ R}; 0 6 ‘ 6 s−1. All elements of Bi‘ ; 0 6 ‘ 6 s−1; i ∈ I (‘); are in H e ; thus the blocks of F‘∗ are mutually disjoint since distinct elements of R belong to di%erent cosets of He and Bil ’s are mutually disjoint. Noting that the list of di%erences arising from the block rBi‘ is r·MBi‘ ;
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we have that MF‘∗ = r∈R i∈I (‘) r · MBi‘ = R · MF‘ = R · H e = (GF(q n ) − {0}). Thus every F‘∗ ; 0 6 ‘ 6 s − 1; is a (G(q n ); k; )-DDF. Meanwhile 06‘6s−1 B∈F∗ B = ‘ n ‘ e 06‘6s−1 i∈I (‘) r∈R rBi = R · (GF(q) − {0}) = R · H = GF(q ) − {0}; that is; the base ∗ blocks of F‘∗ ; 0 6 ‘ 6 s − 1; form a partition of GF(q n ) − {0}. Hence {F0∗ ; : : : ; Fs−1 } form a (G(q n ); k; )-CDDF. We note that Lemma 3.2 can also be deduced from Theorem 2.3 and Lemma 2.1. The next assertion can be checked easily, so we omit its proof. Theorem 3.3. Let q = 6t + 1 be a prime power. Then {F0 ; F1 }; where Fj = {!i+jt H 2t : 0 6 i 6 t − 1}; j = 0; 1; forms a (G(q); 3; 1)-CDDF. It can also be easily seen that the necessary condition for the existence of such a (G(q); 3; 1)-CDDF is q = 6t + 1 for some t ∈ N. By applying Theorem 2.3 with Lemma 3.2 and Theorem 3.3, we get the following result. Corollary 3.4. There exists a (3u; 3; 1)-CDDF; where the prime power factorization of u has the form u = uini where ui = 25; 55; or a prime congruent to 1 module 6. Now we use a result of Buratti (1995a) to establish a necessary and suOcient condition for the existence of a special (G(q); 5; 1)-CDDF. Lemma 3.5. (Buratti; 1995a). Let q = 20t + 1 be a prime power; 2e the highest power of 2 in t; and ' a primitive 5th root of unity in the ;eld GF(q). Then the necessary and su
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Next we consider the suOciency. Assume that the condition, which is described in Lemma 3.5, holds. Then, there is an integer 0 6 f 6 e such that ' + 1 is a 2f th power but not a 2f+1 th power in GF(q). Consider the following four subsets: X0 = {!2
f+1
X1 = {!2
i+j
f+1
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f };
i+2f +j
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f };
X2 = {!2t+2
f+1
i+j
X3 = {!2t+2
f+1
i+2f +j
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f }; : 0 6 i ¡ t=2f ; 0 6 j ¡ 2f }:
Buratti (1995a) proved that the family F0 = {xH 4t : x ∈ X0 } is a (G(q); 5; 1)-DF. Evidently, the base blocks of F0 are mutually disjoint. Similarly, prove that we can Fi = {xH 4t : x ∈ Xi }; i = 1; 2; 3, are also (G(q); 5; 1)-DDFs. Since 06i63 B∈Fi B form a partition of GF(q) − {0}, we know that {Fi : i = 0; 1; 2; 3}, where Fi = {xH 4t : x ∈ Xi }, forms a (G(q); 5; 1)-CDDF. Buratti (1995a) veriHed that the primes q = 20t + 1 ¡ 104 satisfying the condition mentioned above are the following: 41; 61; 241; 281; 401; 421; 601; 641; 661; 701; 761; 821; 881; 1361; 2441; 2801; 3001; 3121; 3881; 4001; 4201; 4561; 5281; 5441; 6481; 6521; 6761; 6841; 6961; 7841; 8161; 8641; 9001; 9521: We continue to investigate the case for odd block size ¿ 5. Theorem 3.7. Let q ≡ 1 (mod k(k − 1)) be a prime power; k = 2m + 1; ' a primitive kth root of unity in GF(q); and assume that {' − 1; : : : ; 'm − 1} is a system of distinct representatives for the cyclotomic classes Hm . Then there exists a (G(q); k; 1)-CDDF. Proof. Assume q − 1 = tk(k − 1) = 2tm(2m + 1). Let B = {1; '; : : : ; 'k−1 }. Then ' = !2tm . Put F0 = {B · !im : i = 0; : : : ; t − 1}; F1 = {B · !im+1 : i = 0; : : : ; t − 1}; .. . Fm−1 = {B · !im+m−1 : i = 0; : : : ; t − 1}; Fm = {B · !tm+im : i = 0; : : : ; t − 1}; Fm+1 = {B · !tm+im+1 : i = 0; : : : ; t − 1}; .. . F2m−1 = {B · !tm+im+m−1 : i = 0; : : : ; t − 1}:
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Wilson (1972) proved that if {' − 1; : : : ; 'm − 1} is an SDRC(Hm ) then F0 is a (G(q); k; 1)-DF; whose base blocks are evidently mutually disjoint. Similarly; we can prove that F1 ; : : : ; F2m−1are also (G(q); k; 1)-DFs with mutually disjoint base blocks. It can be checked that 06j62m−1 B∈Fj B form a partition of GF(q) − {0}. Hence {F0 ; : : : ; F2m−1 } forms a (G(q); k; 1)-CDDF. Wilson (1972) used a computer to check the validity of the hypothesis of Theorem 3.7 for several values of k and q. He succeeded in Hnding such a primitive kth root of unity in GF(q) for q = 337; 421; 463; 883; 1723; 3067; 3319 when k = 7, for q = 73; 1153; 1873; 2017 when k = 9, and for q = 76231 when k = 15. Recently, Buratti (1995b) gave necessary and suOcient conditions for the validity of the hypothesis of Theorem 3.7 for k = 7 and also gave a suOcient condition for k ¿ 7. The interested reader is referred to Buratti (1995b) for details. The above constructions are only applicable to the case when the block size is odd. What can we say about the case when the block size is even? Obviously we can not use the same idea to produce complete sets of disjoint di%erence families with even block size and = 1 in Hnite Helds. Here, we present a direct construction for complete sets of disjoint di%erence families which is applicable irrespective of the parity of the block size. This method of construction is rather di%erent from those described above. Let B = {b0 ; b1 ; : : : ; bk−1 }, where bi ∈ GF(q); 0 6 i 6 k − 1, and q = k(k − 1)t + 1; t ∈ N. We deHne the following k − 1 families of blocks of size k: Fj = {B · !k(k−1)i+kj : 0 6 i 6 t − 1};
0 6 j 6 k − 2:
In order for {F0 ; : : : ; Fk−2 } to be a (G(q); k; 1)-CDDF we need some suOcient conditions on the k-dimensional vector (b0 ; b1 ; : : : ; bk−1 ). Theorem 3.8. Let q = k(k − 1)t + 1 be a prime power. If there exist k elements of GF(q); b0 ; b1 ; : : : ; bk−1 ; satisfying the following conditions; then there exists a (G(q); k; 1)-CDDF. (C1 ) The set .{b0 ; : : : ; bk−1 } of k(k − 1) di&erences among {b0 ; : : : ; bk−1 } forms an SDRC(Hk(k−1) ); (C2 ) (b0 ; : : : ; bk−1 ) · (!kj : 0 6 j 6 k − 2) forms an SDRC(Hk(k−1) ). Condition (C1 ) is required for the di%erences arising from the blocks in Fj ; 0 6 j 6 k − 2, so that every nonzero element of GF(q) can occur precisely once, while Condition (C2 ) is required for the blocks of 06j6k−2 Fj to be a partition of GF(q)−{0}. Immediately we can know that t must be odd. For otherwise (b0 −b1 )=(b1 −b0 ) ∈ H k(k−1) , contradicting (C1 ). We focus our attention on the case when k = 4. Let (b0 ; b1 ; b2 ; b3 ) = (1; x; x2 ; x3 ) for some x ∈ GF(p) − {0; 1}, where p = 12t + 1 is a prime number and t is odd. If there exists an element x ∈ H112 such that 1 + x ∈ H312 ;
1 + x + x2 ∈ H512 ; or
1 + x ∈ H912 ;
1 + x + x2 ∈ H512 ; or
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Table 1 p
!
(b0 ; b1 ; b2 ; b3 )
61 109 181 397
2 6 2 5
(1; 8; 27; 24) (1; 6; 100; 33) (1; 2; 168; 151) (1; 5; 324; 343)
1 + x ∈ H412 ;
1 + x + x2 ∈ H912 ; or
12 1 + x ∈ H10 ;
1 + x + x2 ∈ H912 ;
then it can be easily checked that (1; x; x2 ; x3 ) satisHes our Conditions (C1 ) and (C2 ). By using Weil’s theorem on character sums, and by the aid of a computer, Lam and Miao (1999) found such an element x of GF(p) satisfying one of the above systems for every prime p = 12t + 1; t odd, except for p = 13; 61; 109; 181 and 397. For the remaining Hve cases, we ran a computer program to Hnd (b0 ; b1 ; b2 ; b3 ) which satisfy our Conditions (C1 ) and (C2 ). We found solutions for all these remaining cases except for p=13. We list p; ! and (b0 ; b1 ; b2 ; b3 ) in Table 1, where ! is the primitive element of GF(p) we used in our computation. We did not Hnd any suitable (b0 ; b1 ; b2 ; b3 ) for p = 13 by exhaustive search. In fact, the following 4-subsets are all the possible di%erence families with k = 4 and = 1 over G(13). {0; 1; 3; 9}; {0; 1; 4; 6}; {0; 1; 5; 11}; {0; 1; 8; 10}; {0; 2; 3; 7}; {0; 2; 5; 6}; {0; 2; 8; 12}; {0; 2; 9; 10}; {0; 3; 4; 11}; {0; 3; 5; 12}; {0; 4; 5; 7}; {0; 4; 10; 12}; {0; 6; 8; 9}; {0; 6; 10; 11}; {0; 7; 8; 11}; {0; 7; 9; 12}; {1; 2; 4; 10}; {1; 2; 5; 7}; {1; 2; 6; 12}; {1; 2; 9; 11}; {1; 3; 4; 8}; {1; 3; 6; 7}; {1; 3; 10; 11}; {1; 4; 5; 12}; {1; 5; 6; 8}; {1; 7; 9; 10}; {1; 7; 11; 12}; {1; 8; 9; 12}; {2; 3; 5; 11}; {2; 3; 6; 8}; {2; 3; 10; 12}; {2; 4; 5; 9}; {2; 4; 7; 8}; {2; 4; 11; 12}; {2; 6; 7; 9}; {2; 8; 10; 11}; {3; 4; 6; 12}; {3; 4; 7; 9}; {3; 5; 6; 10}; {3; 5; 8; 9}; {3; 7; 8; 10}; {3; 9; 11; 12}; {4; 5; 8; 10}; {4; 6; 7; 11}; {4; 6; 9; 10}; {4; 8; 9; 11}; {5; 6; 9; 11}; {5; 7; 8; 12}; {5; 7; 10; 11}; {5; 9; 10; 12}; {6; 7; 10; 12}; {6; 8; 11; 12}:
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It can be easily checked that no three of them can be mutually disjoint. So we can assert that there is no (G(13); 4; 1)-CDDF. Furthermore, since there is no other group of order 13, we can even say that there is no complete set of disjoint di%erence families of block size k = 4 and index = 1 over any group of order 13. Summarizing the above, we can get the following result. Theorem 3.9. There exists a (G(p); 4; 1)-CDDF whenever p = 12t + 1 is a prime number and t is odd; except for p = 13. We do not know, by now, whether there is a similar result for every even block size k ¿ 6. Weil’s theorem could provide a bound p0 (k) for each even block size k ¿ 6 such that for any prime p = k(k − 1)t + 1, where t is odd and p ¿ p0 (k), there exists a (G(q); k; 1)-CDDF. A computer could also be used to Hnd a proper element x ∈ GF(q) for each small prime p = k(k − 1)t + 1; t odd, to guarantee the existence of a (G(q); k; 1)-CDDF for even k ¿ 6. But it seems impractical for us at this moment to ask a computer to Hnd such an element x ∈ GF(q) for all prime p ¡ p0 (k) with p = k(k − 1)t + 1 and t is odd for any even block size k ¿ 6. A new method is desired for a complete solution to this problem. Sometimes, the properties of Hnite rings can also be used to produce various combinatorial structures. For example, complete sets of disjoint di%erence families can also be constructed directly from Hnite rings. Let R be a commutative ring of odd order admitting a unit ' of order k, where k is an odd integer, such that {' − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Consider the relation ∼ deHned in R − {0} by x∼y
if and only if
y = 's x for some s ∈ {0; 1; : : : ; k − 1}:
It is easy to see that ∼ is an equivalence relation whose equivalence class is the set r' = {r; r'; : : : ; r'k−1 }; r ∈ R − {0}. Since {' − 1; : : : ; '(k−1)=2 − 1} is a set of units, each of these sets has actually size k. Hence a system of distinct representatives for the equivalence classes of ∼ has cardinality (|R| − 1)=k. Let X be such a system. If there exists a set X0 such that X =±X0 , then {X0 '; −X0 '} forms a (G; k; (k −1)=2)-CDDF, where G is the additive group of R. This can be checked as follows. The di%erences arising from '; M', is ±' · (' − 1; : : : ; '(k−1)=2 − 1). Then M(X0 ') = X0 · M' = X0 · ±'·('−1; : : : ; '(k−1)=2 −1)=±X0 ·'·('−1; : : : ; '(k−1)=2 −1)=X ·'·('−1; : : : ; '(k−1)=2 −1) = (G − {0}) · (' − 1; : : : ; '(k−1)=2 − 1) = ((k − 1)=2)(G − {0}); M(−X0 ') = −MX0 · ' = ((k − 1)=2)(G − {0}), and (X0 · ') ∪ (−X0 · ') = ±X0 · ' = X · ' = G − {0}. Lemma 3.10. Let R be a commutative ring of odd order admitting a unit of order k; where k is an odd integer; such that {' − 1; : : : ; 'k−1 − 1} is a set of units of R. Let X be a system of distinct representatives for the equivalence classes of ∼ de;ned above. Then; there exists a set X0 such that X = ±X0 . Proof. For any x ∈ X we claim that x' ∩ (−x') = ∅. Otherwise there would exist x0 ∈ X and s ∈ {0; 1; : : : ; k −1} such that −x0 =x0 's . Since ' is a unit of order k; −x0 'k = x0 's ; which means −x0 =x0 'k−s . So we know that x0 'k−s =x0 's ; which implies x0 ('k−2s −1) = 0. If k − 2s ≡ 0 (mod k); then by our hypothesis; 'k−2s − 1 is a unit; and then x0 = 0;
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which is impossible. So we must have that k − 2s ≡ 0 (mod k); i.e. 2s ≡ 0 (mod k). Since k is odd; s ≡ 0 (mod k); and thus s=0. This means −x0 =x0 ; which is impossible too since |R| is odd. Therefore, we can separate X into two sets X0 and −X0 so that x ∈ X0 if and only if −x ∈ (−X0 ). Here we note that the additive group G of the ring R contains no nonzero element which is its own additive inverse. As a summary, we have the following result. Theorem 3.11. Let R be a commutative ring of odd order admitting a unit ' of order k; where k is an odd integer; such that {' − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Then there exists a (G; k; (k − 1)=2)-CDDF; where G is the additive group of R. For example, we have the following two consequences. Corollary 3.12. Let v and k be odd integers such that p ≡ 1 (mod k) holds for each prime p in v. Then there exists a (Zv ; k; (k − 1)=2)-CDDF. Proof. Let v = i∈I pini be the prime power factorization of v. Then by the Chinese Remainder Theorem; we can identify Zv with the ring R = ⊕i∈I Zpini . Buratti (1997) described a unit ' of order k in R such that {' − 1; '2 − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Then the assertion follows from Theorem 3.11. Corollary 3.13. Let v and k be odd integers and v = i∈I pini the prime power factorization of v such that pini ≡ 1 (mod k) for each i ∈ I . Then there exists a (G(v); k; (k − 1)=2)-CDDF; where G(v) is the additive group of the Galois ring GR(v) of order v. Proof. Buratti (1997) also described a unit ' of order k in GR(v) such that {' − 1; '2 − 1; : : : ; '(k−1)=2 − 1} is a set of units of GR(v). Then the assertion follows from Theorem 3.11. In order to clarify the above results, let us see the following example. Consider the Galois ring Z91 . It admits a unit 9 of order 3 such that 9 − 1 = 8 is still a unit. Then the two di%erence families described below form a (Z91 ; 3; 1)-CDDF. {{{1; 9; 81}; {2; 18; 71}; {3; 27; 61}; {4; 36; 51}; {5; 45; 41}; {6; 54; 31}; {7; 63; 21}; {8; 72; 11}; {12; 17; 62}; {13; 26; 52} {14; 35; 42}; {15; 44; 32}; {16; 53; 22}; {23; 25; 43}; {24; 34; 33}} ; {{10; 90; 82}; {20; 89; 73}; {30; 88; 64}; {40; 87; 55}; {46; 50; 86}; {37; 60; 85}; {28; 70; 84}; {19; 80; 83};
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{29; 79; 74} {39; 78; 65}; {49; 77; 56}; {47; 59; 76}; {38; 69; 75}; {48; 68; 66} {67; 57; 58}}} : 4. An application to one-factorizations of complete graphs In the remainder of this paper, we mention two applications of complete sets of disjoint di%erence families. First, we describe an application to the construction of a special kind of one-factorizations of complete graphs. Recall that a complete graph Kn is a graph on n vertices such that every pair of vertices are adjacent, that is, every pair of vertices is connected by an edge. A one-factor in a complete graph Kn is a set of edges in which every vertex appears precisely once. In other words, it is a regular spanning subgraph of degree 1. A one-factorization of Kn is a way of partitioning the edge set of Kn into one-factors. Let G be an abelian group of order 2m + 1, and S be a (G; 2; 1)-CDDF. Then S is in fact a family of unordered pairs {{xi ; yi }: 1 6 i 6 m} which satisHes the following two properties: (P1) {xi : 1 6 i 6 m} ∪ {yi : 1 6 i 6 m} = G − {0}, (P2) {±(xi − yi ): 1 6 i 6 m} = G − {0}. Historically, such family of unordered pairs is called a starter in G, which was introduced by Stanton and Mullin (1968) for the direct construction of Room squares. Later on, starters were also proved to be useful in the construction of one-factorizations of complete graphs, as the following shows. Theorem 4.1. If there is a starter S in an abelian group G of order 2m + 1; then the 2m + 1 factors {∞; x} ∪ {S + x}; where x ranges through G; form a one-factorization of K2(m+1) based on {∞} ∪ G. This construction can be found, for example in Wallis (1992). Here we describe another construction of a one-factorization of K2(2m+1) , which is also generated by a starter. Let S be a starter in an abelian group G of order 2m + 1. Then |S| = m. Let the complete graph K2(2m+1) be based on Z2 ⊕G. Then it can be checked that the edge set developed modulo (Z2 ; G) from {{(0; xi ); (0; yi )}: {xi ; yi } ∈ S}∪{{(0; xi ); (1; yi )}: {xi ; yi } ∈ S} and modulo (−; G) from {(0; 0); (1; 0)} is the edge set of this K2(2m+1) , while FI (S) = {{(0; xi ); (0; yi )}: {xi ; yi } ∈ S} ∪ {{(1; xi ); (1; yi )}: {xi ; yi } ∈ S} ∪ {(0; 0); (1; 0)} is a one-factor. Develop it modulo (−; G) we get 2m + 1 one-factors FI (S) + (−; g), where g ∈ G. Also for each pair {xi ; yi } ∈ S; 1 6 i 6 m; FII (S; i) = {{(0; xi ); (1; yi )} mod(−; G): {xi ; yi } ∈ S} is a one-factor. Develop it modulo (Z2 ; −) we get 2 one-factors FII (S; i) + (z; −), where z ∈ Z2 . Altogether we get 4m + 1 one-factors, which form a one-factorization of the K2(2m+1) based on Z2 ⊕ G. Obviously this one-factorization is invariant under the action of Z2 ⊕ G, with one factor orbit of length 2m + 1, and m factor orbits of length 2.
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Theorem 4.2. If there is a starter in an abelian group G of order 2m + 1; then there is a one-factorization of the K2(2m+1) based on Z2 ⊕G which is invariant under Z2 ⊕G. Note that Theorem 4.2 was Hrst proved for the case when G is a cyclic group of order 2m + 1, that is, G = Z2m+1 , by Hartman and Rosa (1985). We investigate further properties on the intersections of such kind of onefactorizations. Let S = {{si ; ti }: 1 6 i 6 m} and T = {{ui ; vi }: 1 6 i 6 m} be two starters in G. Without loss of generality, we may assume that si − ti = ui − vi for all i; 1 6 i 6 m. If ui −si =uj −sj can imply i =j, and if ui = si for all i; 1 6 i 6 m, then S and T are said to be orthogonal. Two orthogonal starters S and T can produce two one-factorizations of the K2(2m+1) based on Z2 ⊕G which are invariant under Z2 ⊕G. Let hi =si −ui , for all i; 1 6 i 6 m. Then, by the deHnition of orthogonal starters, we can get the following result on the intersection problem of such two one-factorizations. (FI (S) + (−; g)) ∩ (FI (T ) + (−; g) + (−; hi )) = {{(0; si ); (0; ti )}; {(1; si ); (1; ti )}} for all g ∈ G and for all hi ; 1 6 i 6 m; (FI (S) + (−; g)) ∩ (FI (T ) + (−; g)) = {{(0; 0); (1; 0)}}
for all g ∈ G;
(FI (S) + (−; g)) ∩ (FI (T ) + (−; g) + (−; h)) =∅
for all g ∈ G and for all h ∈ G − {0} ∪ {hi : 1 6 i 6 m};
(FI (S) + (−; g)) ∩ (FII (T; i) + (z; −)) =∅
for all g ∈ G; all z ∈ Z2 ; and all i; 1 6 i 6 m;
(FI (T ) + (−; g)) ∩ (FII (S; i) + (z; −)) =∅
for all g ∈ G; all z ∈ Z2 ; and all i; 1 6 i 6 m;
(FII (S; i) + (z; −)) ∩ (FII (T; i) + (z; −)) = FII (S; i) + (z; −) = FII (T; i) + (z; −)
for all z ∈ Z2 ;
(FII (S; i) + (z; −)) ∩ (FII (T; i) + (z + 1; −)) =∅
for all z ∈ Z2 and all i; 1 6 i 6 m;
(FII (S; i) + (z1 ; −)) ∩ (FII (T; j) + (z2 ; −)) =∅
for all i = j; 1 6 i; j 6 m; all z1 ∈ Z2 and all z2 ∈ Z2 :
The interested reader is referred to Lindner and Wallis (1982) for a detailed discussion on the intersection problem of two one-factorizations.
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5. An application to cyclically resolvable cyclic Steiner systems Let V be a set of v elements, and B a collection of k-subsets of V called blocks. A balanced incomplete block design (BIBD), denoted by (v; k; )-BIBD, is a pair (V; B) such that every pair of distinct elements of V is contained in precisely blocks of B. A BIBD with = 1 is called a Steiner 2-system, while a Steiner 2-system with block size k = 3 is a Steiner triple system. An automorphism of a BIBD (V; B) is a permutation on V whose induced mapping from B to B is a permutation too. The set of all automorphisms of a BIBD forms a group, the full automorphism group. Any of its subgroups is called an automorphism group of the BIBD. A (v; k; )-BIBD admitting a cyclic and element-regular automorphism group is called a cyclic (v; k; )-BIBD. For a cyclic BIBD (V; B), the set V of elements can be identiHed with Zv . In this case, the BIBD has an automorphism : i → i + 1 (mod v). If a block orbit is of length v, then this block orbit is said to be full, otherwise short. The block orbit which contains the following block is called a regular short block orbit (v=k)Zk = {0; v=k; : : : ; (k − 1)v=k}: A resolution of a BIBD (V; B) is a partition of B into resolution classes each of which is a partition of V. If such a resolution exists, then (V; B) is said to be resolvable. Let G and R be an automorphism group and a resolution of a BIBD (V; B), respectively. If R is Hxed by G, i.e. if Rig ∈ R for all (g; Ri ) ∈ G × R, we say that R is G-invariant. In this case, we also say that (V; B) is G-invariantly resolvable. Trivially, “{1v }-invariantly resolvable” simply means “resolvable”, while a Zv invariantly resolvable cyclic (v; k; )-BIBD is commonly called cyclically resolvable cyclic (v; k; )-BIBD, and denoted by (v; k; )-CRCB. It is clear now that the one-factorization of K2(2m+1) we constructed in Section 4 is in fact a Z2 ⊕ G-invariantly resolvable (2(2m + 1); 2; 1)-BIBD. In this section, we will consider a special case of such BIBDs with k = 3 and = 1, while G is a cyclic group, i.e. cyclically resolvable cyclic Steiner triple systems. Of course, we can adapt it for non-cyclic cases. Cyclically resolvable cyclic Steiner 2-systems and related structures have been investigated by several authors, see for example (Genma et al., 1997; Kageyama and Miao, 1998; Lam and Miao, 1999; Mishima and Jimbo, 1997). Mishima and Jimbo (1997) classiHed cyclically resolvable cyclic Steiner 2-systems into three types (T1), (T2) and (T3), according to their relation with cyclic quasiframes, cyclic semiframes, or cyclically resolvable cyclic group divisible designs. In particular, in type (T2), each block in the regular short block orbit is in a distinct resolution class. The reader is referred to Mishima and Jimbo (1997) for their formal deHnitions. Before we describe our method of construction, we need some necessary condition on the number of elements in a cyclically resolvable cyclic Steiner triple system of type (T2). An obvious necessary condition for the existence of a resolvable cyclic Steiner triple system with number of elements v is v ≡ 3 (mod 6). In the case of a cyclically resolvable cyclic Steiner triple system of type (T2), a stronger condition is needed.
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Lemma 5.1. (Genma et al.; 1997). For a (3u; 3; 1)-CRCB of type (T 2); u ≡ 1 (mod 6) holds. So we can assume that u=6t+1 for some integer t ∈ N∪{0}. Since gcd(3; u)=1, we know that Z3u Z3 ⊕Zu . In this section, instead of using Z3u as the set of elements and : i → i + 1 (mod 3u) as the automorphism, we will use Z3 ⊕ Zu as the set of elements and : (i; j) → (i; j) + (1; 1) (mod (3; u)) as the automorphism. With this representation, it is easier to see the structure of the CRCB. Now we provide our direct construction for (3u; 3; 1)-CRCB of type (T2) based on the set of elements V = Z3 ⊕ Zu , where u = 6t + 1 for some t ∈ N ∪ {0}. Theorem 5.2. If there exists a (Z6t+1 ; 3; 1)-CDDF then there exists a (3(6t + 1); 3; 1)CRCB of type (T 2). Proof. We Hrst construct the following 3t + 1 blocks: A = {(0; 0); (1; 0); (2; 0)}; Bi = {(0; bi0 ); (0; bi1 ); (0; bi2 )};
1 6 i 6 t;
Ci = {(0; ci0 ); (1; ci1 ); (2; ci2 )};
1 6 i 6 t;
Di = {(0; di0 ); (1; di1 ); (2; di2 )};
1 6 i 6 t:
We try to Hnd some suOcient conditions on bij ; cij ; dij ; 1 6 i 6 t; 0 6 j 6 2; so that we could get a (3u; 3; 1)-CRCB of type (T2); where V = Z3 ⊕ Z6t+1 ; B = {A (mod (−; 6t + 1))} ∪ {Bi (mod (3; 6t + 1)): 1 6 i 6 t} ∪ {Ci (mod (3; 6t + 1)): 1 6 i 6 t} ∪ {Di (mod (3; 6t + 1)): 1 6 i 6 t}; and a base resolution class of a resolution class orbit modulo (−; 6t + 1) is {A} ∪ {Bi (mod (3; −)): 1 6 i 6 t} ∪ {Ci (mod (3; −)): 1 6 i 6 t}; while a base resolution class of a resolution class orbit modulo (3; −) is {Di (mod (−; 6t + 1)): 1 6 i 6 t}: The conditions are described in the following: (C1) {±(bi0 − bi1 ); ±(bi0 − bi2 ); ±(bi1 − bi2 ): 1 6 i 6 t} = Z6t+1 − {0}; (C2) {ci0 − ci1 ; ci1 − ci2 ; ci2 − ci0 ; di0 − di1 ; di1 − di2 ; di2 − di0 } = Z6t+1 − {0}; (C3) {bij ; cij : 1 6 i 6 t; 0 6 j 6 2} = Z6t+1 − {0}.
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Condition (C1) is required for the pure di%erences arising from the blocks in B; Condition (C2) is required for the mixed di%erences arising from the blocks in B; and Condition (C3) is required for the resolution class which contains a block from the regular short block orbit A = {A (mod (−; 6t + 1))}. If we take di0 = ci0 ; di1 = ci2 and di2 = ci1 , then {ci0 − ci1 ; ci1 − ci2 ; ci2 − ci0 ; di0 − di1 ; di1 − di2 ; di2 − di0 } = {±(ci0 − ci1 ); ±(ci1 − ci2 ); ±(ci0 − ci2 )}. That is, Condition (C2) becomes Condition (C2 ) (C2 ) {±(ci0 − ci1 ); ±(ci1 − ci2 ); ±(ci0 − ci2 )} = Z6t+1 − {0}: Therefore, what we need is a doubly disjoint di%erence family (Z6t+1 ; 3; 1)-CDDF, where F0 = {{bi0 ; bi1 ; bi2 }: 1 6 i 6 t} and F1 = {{ci0 ; ci1 ; ci2 }: 1 6 i 6 t}. It completes the proof. As an immediate consequence, we have the following result. Corollary 5.3. There exists a (3u; 3; 1)-CRCB of type (T 2); where the prime power factorization of u has the form u = uini where ui = 25; 55; or a prime congruent to 1 modulo 6. Proof. Such a (u; 3; 1)-CDDF can be found in Corollary 3.4. Acknowledgements The authors would like to thank the referees for their valuable suggestions in revising this paper. References Abel, R.J.R., 1996. Di%erence families. In: Colbourn, C.J., Dinitz, J.H. (Eds.), The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 270–287. Buratti, M., 1995a. Improving two theorems of Bose on di%erence families. J. Combin. Des. 3, 15–24. Buratti, M., 1995b. On simple radical di%erence families. J. Combin. Des. 3, 161–168. Buratti, M., 1997. From a (G; k; 1) to a (Ck ⊕ G; k; 1) di%erence family. Des. Codes Cryptogr. 11, 5–9. Colbourn, M.J., Colbourn, C.J., 1984. Recursive constructions for cyclic block designs. J. Statist. Plann. Inference 10, 97–103. Colbourn, C.J., de Launey, W., 1996. Di%erence matrices. In: Colbourn, C.J., Dinitz, J.H. (Eds.), CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 287–297. Dinitz, J.H., 1996. Starters. In: Colbourn, C.J., Dinitz, J.H. (Eds.), CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 467–473. Dinitz, J.H., Rodney, P., 1997. Disjoint di%erence families with block size 3. Utilitas Math. 52, 153–160. Dinitz, J., Shalaby, N., Block disjoint di%erence families for Steiner triple systems: v ≡ 3 mod 6. J. Statist. Plann. Inference, Special issue dedicated to the memory of Professor Sumiyashu Yamamoto. Furino, S., Miao, Y., Yin, J., 1996. Frames and Resolvable Designs: Uses, Constructions, and Existence. CRC Press, Boca Raton, FL. Genma, M., Mishima, M., Jimbo, M., 1997. Cyclic resolvability of cyclic Steiner 2-designs. J. Combin. Des. 5, 177–187.
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Hartman, A., Rosa, A., 1985. Cyclic one-factorization of the complete graph. European J. Combin. 6, 45–48. Jimbo, M., 1993. Recursive constructions for cyclic BIB designs and their generalizations. Discrete Math. 116, 79–95. Jungnickel, D., 1978. Composition theorems for di%erence families and regular planes. Discrete Math. 23, 151–158. Kageyama, S., Miao, Y., 1998. A construction for resolvable designs and its generalizations. Graphs Combin. 14, 11–24. Lam, C., Miao, Y., 1999. On cyclically resolvable cyclic Steiner 2-designs, J. Combin. Theory, Ser. A 85, 194–207. Lindner, C.C., Wallis, W.D., 1982. A note on one-factorizations having a prescribed number of edges in common. Ann. Discrete Math. 12, 203–209. Mishima, M., Jimbo, M., 1997. Some types of cyclically resolvable cyclic Steiner 2-designs. Congr. Numer. 123, 193–203. Stanton, R.G., Mullin, R.C., 1968. Construction of room squares. Ann. Math. Statist. 39, 1540–1548. Wallis, W.D., 1992. One-factorizations of complete graphs. In: Dinitz, J.H., Stinson, D.R. (Eds.), Contemporary Design Theory: A Collection of Surveys. Wiley, New York, pp. 593–631. Wilson, R.M., 1972. Cyclotomy and di%erence families in elementary abelian groups. J. Number Theory 4, 17–47.
www.elsevier.com/locate/jspi
Complete sets of disjoint di%erence families and their applications R. Fuji-Haraa; ∗ , Y. Miaoa , S. Shinoharab a Institute
b Doctoral
of Policy and Planning Sciences, University of Tsukuba, Tsukuba 305-8573, Japan Program in Policy and Planning Sciences, University of Tsukuba, Tsukuba 305-8573, Japan Received 2 December 1998; accepted 15 February 1999 Dedicated to the memory of Professor Sumiyasu Yamamoto
Abstract Let G be an abelian group. A collection of (G; k; ) disjoint di%erence families, {F0 ; F1 ; : : : ; Fs−1 }, is a complete set of disjoint di%erence families if 06i6s−1 B∈Fi B form a partition of G − {0}. In this paper, several construction methods are provided for complete sets of disjoint di%erence families. Applications to one-factorizations of complete graphs and to cyclic 2002 Elsevier Science B.V. cally resolvable cyclic Steiner triple systems are also described. All rights reserved. Keywords: Complete set of disjoint families; Cyclically resolvable cyclic Steiner system; One-factorization of a complete graph
1. Introduction Let G be an abelian group of order v; k an integer satisfying 2 6 k 6 v, and a positive integer. A (G; k; ) di&erence family, denoted by (G; k; )-DF, is a collection F = {Bi : i ∈ I } of k-subsets of G, called base blocks, such that any nonzero element of G can be represented in precisely ways as a di%erence of two elements lying in some base blocks of F. The number of base blocks of a (G; k; )-DF is obviously (|G|−1)=(k(k −1)), and hence a necessary condition for the existence of a (G; k; )-DF is that (|G| − 1) ≡ 0 (mod k(k − 1)) holds. If the base blocks of a (G; k; )-DF are mutually disjoint, then this (G; k; )-DF is said to be disjoint, and denoted by (G; k; )-DDF. (G; k; )-DFs and (G; k; )-DDFs have been investigated intensively, see ∗
Corresponding author. E-mail addresses: [email protected] (R. Fuji-Hara), [email protected] (Y. Miao), [email protected] (S. Shinohara). c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 2 0 5 - 7
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for example (Abel, 1996; Dinitz and Rodney, 1997; Dinitz and Shalaby, to appear; Furino et al., 1996; Wilson, 1972). Let {F0; F1 ; : : : ; Fs−1 } be a collection of (G; k; ) disjoint di%erence families. If 06i6s−1 B∈Fi B form a partition of G − {0}, then the collection of (G; k; )-DDFs {F0 ; F1 ; : : : ; Fs−1 } is called a complete set of disjoint di%erence families and denoted by (G; k; )-CDDF, whereeach Fi ; 0 6 i 6 s − 1, is the component of the (G; k; )-CDDF. Clearly, {B: B ∈ 06i6s−1 Fi } forms a (G; k; s)-DDF, while the number s of component disjoint di%erence families therein is (k − 1)=. When = k − 1, a (G; k; )-CDDF is just a (G; k; )-DDF. Complete sets of disjoint di%erence families are of interest in their own right, as well as having applications in the construction of other types of combinatorial structures. Starters in an abelian group G are in fact (G; 2; 1)-CDDFs, which have been used to construct Room squares, round robin tournaments, one-rotational one-factorizations of complete graphs, etc., see for example Dinitz (1996). (G; k; k − 1)-CDDFs are just (G; k; k − 1)-DDFs, which have been used to construct resolvable balanced incomplete block designs, see for example Furino et al. (1996). Complete sets of disjoint di%erence families also have new applications which will be described later. In this paper, we present several constructions for complete sets of disjoint di%erence families, mainly with = k − 1. Composition theorems are used to produce new complete sets of disjoint di%erence families from two collections of “small” ones by means of di%erence matrices. A computer is used to construct two individual examples of (Z6t+1 ; 3; 1)-CDDFs, where t = 4 and 9. Some complete sets of disjoint di%erence families over Hnite Helds and Hnite rings are constructed directly by exploiting some properties of Hnite Helds and Hnite rings. Two applications of complete sets of disjoint di%erence families are also mentioned: a (G; 2; 1)-CDDF is used to construct a one-factorization of the complete graph based on Z2 ⊕ G which is invariant under the automorphism group Z2 ⊕ G, while a (G; 3; 1)-CDDF is used to construct a cyclically resolvable cyclic Steiner triple system of order 3|G|. 2. Composition theorems Before describing any method of construction, we Hrst make a simple observation on complete sets of disjoint di%erence families. Let G be an abelian group. If there is a (G; k; )-CDDF, where is a factor of k −1, then by combining its n (1 6 n 6 s) component (G; k; )-DDFs into one (G; k; n)-DDF, we can get a (G; k; n)-CDDF, provided that n is still a factor of k − 1. Jungnickel (1978), Jimbo (1993), and Kageyama and Miao (1998) presented several composition theorems for di%erence families with or without other properties. The following are their variations for complete sets of disjoint di%erence families. Here a (G; k; ) di&erence matrix, denoted by (G; k; )-DM, is a k × |G| matrix (mij ) with entries from an abelian group G such that the list of di%erences (mis −mjs : 0 6 s 6 |G|−1) contains each element of G precisely times, where i = j and 0 6 i; j 6 k −1. Di%erence matrices have been studied widely, see for example, (Colbourn and de Launey (1996)). The following simple assertion can be found in Colbourn and Colbourn (1984).
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Lemma 2.1. (Colbourn and Colbourn; 1984). Let ij ≡ ij (mod w) for i=0; 1; : : : ; k −1; and j = 0; 1; : : : ; w − 1; where gcd((k − 1)!; w) = 1. Then = (ij ) is a (Zw ; k; 1)-DM. In particular; if w is a prime number; then there exists a (Zw ; k; 1)-DM for any integer k (6 w). Di%erence matrices are useful in the construction of complete sets of disjoint di%erence families. Theorem 2.2. Let G1 and G2 be two abelian groups. If there exist a (G1 ; k; )-CDDF; a (G2 ; k; )-CDDF and a (G2 ; k + 1; 1)-DM; then there exists a (G1 ⊕ G2 ; k; )-CDDF. Proof. First of all; we note that the property of the di%erence matrix is conserved even if we add an element to any columns or any rows. Thus; without loss of generality; we may assume that in a di%erence matrix; the elements in the Hrst row and in the Hrst column are all 0’s. l Let {D0 ; D1 ; : : : ; Ds−1 } be the (G1 ; k; )-CDDF with Dl = {D0l ; D1l ; : : : ; Dr−1 }; Dil = {dli0 ; : : : ; dli; k−1 }; 0 6 l 6 s − 1; 0 6 i 6 r − 1, where r = (|G1 | − 1)=(k(k − 1)); l }; s=(k −1)=. Let {E0 ; E1 ; : : : ; Es−1 } be the (G2 ; k; )-CDDF with El ={E0l ; E1l ; : : : ; Et−1 l l l Ej = {ej0 ; : : : ; ej; k−1 }; 0 6 l 6 s − 1; 0 6 j 6 t − 1, where t = (|G2 | − 1)=(k(k − 1)). Further, let A = (aij ); 0 6 i 6 k − 1; 0 6 j 6 |G2 | − 1 be the (G2 ; k; 1)-DM derived from the (G2 ; k + 1; 1)-DM by removing its Hrst row. Put Fl = {Fijl : 0 6 i 6 r − 1; 0 6 j 6 |G2 | − 1} with Fijl = {(dli0 ; a0j ); : : : ; (dli; k−1 ; ak−1; j )};
El = {Eil : 0 6 i 6 t − 1}
l with Eil = {(0; ei0 ); : : : ; (0; ei;l k−1 )}
and let Fl be the union of Fl and El . Kageyama and Miao (1998) proved that Fl is a (G1 ⊕ G2 ; k; )-DDF. So we only need to prove that all the base blocks of Fl ; l = 0; : : : ; s − 1, form a partition of G1 ⊕ G2 − {(0; 0)}. We are to prove that each element is distinct from others. If (dli m ; am j ) = (dlim ; amj ) then dli m = dlim and am j = amj . By the deHnition of a complete set of disjoint di%erence families, we can conclude that l = l; i = i and m = m. Then, by the property of the di%erence matrix A, we should have j = j . The cases of (dlim ; amj ) = (0; eil m ) are impossible since dlim = 0. l l If (0; eim = eil m . By the deHnition of a complete set of disjoint ) = (0; eil m ), then eim di%erence families, l = l ; i = i and m = m should also hold. Since these elements are all not (0; 0), and are just all the elements of Fl ; l = 0; : : : ; s − 1, they form a partition of G1 ⊕ G2 − {(0; 0)}. The construction method of Theorem 2.2 does not contain the construction of complete sets of cyclic disjoint di%erence families (i.e., the abelian group G is cyclic). However, the structures of groups are important to our later applications. So we also need the following construction.
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Theorem 2.3. Let G2 be a subgroup of an abelian group G such that G=G2 is isomorphic to an abelian group G1 of order being not k. If there exist a (G1 ; k; )CDDF; a (G2 ; k; )-CDDF and a (G2 ; k + 1; 1)-DM; then there exists a (G; k; )CDDF. This theorem can be proved in a similar way to that of Theorem 2.2, while the construction of corresponding (G; k; )-DDF can be found in Theorem 6.2 of Kageyama and Miao (1998). We also note that Theorem 2.3 can be extended to the case of a non-abelian group. Of course, the notions of a complete set of disjoint di%erence families and a di%erence matrix should be extended simultaneously to the case of a non-abelian group. We will not go further here. 3. Direct constructions In order for the composition theorems described in Section 2 to work, we need several complete sets of “small” disjoint di%erence families to start with. We will describe direct constructions for such kind of di%erence families in this section, most of which are based on Hnite Helds. We Hrst consider those cases where no prime Hnite Held can exist. In such cases, the commonly used technique is to use a computer to search their solutions. We succeeded in Hnding the smallest two cases for k = 3, i.e., a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF. Note that although a Hnite Held of order 25 exists, and 25 ≡ 1 (mod 6), a prime Hnite Held of order 25 does not exist. Lemma 3.1. There exist a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF. Proof. The following two collections of blocks are base blocks of a (Z25 ; 3; 1)-CDDF and a (Z55 ; 3; 1)-CDDF; respectively: {{{5; 12; 10}; {21; 13; 9}; {18; 17; 7}; {3; 6; 22}} ; {{4; 15; 2}; {19; 24; 23}; {1; 16; 8}; {20; 11; 14}}} and {{{6; 27; 47}; {31; 25; 50}; {14; 12; 51}; {53; 45; 48}; {33; 16; 7}; {8; 20; 19}; {54; 30; 26}; {42; 9; 2}; {28; 5; 15}} ; {{34; 36; 21}; {41; 46; 38}; {22; 52; 29}; {35; 4; 24}; {18; 1; 17}; {49; 13; 39}; {3; 44; 40}; {23; 11; 32}; {37; 10; 43}}} :
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We have used the terminology “list” in the deHnition of a di%erence matrix. By a list we mean a collection of elements in which each element occurs nonnegative times. Note that “list” is also called “multiset” sometimes. We use the notation (x1 ; : : : ; x‘ ). The orderis not taken into account in our lists. If Xi ; i = 1; 2; : : : ; t, are lists, then the t notation i=1 Xi is used to denote the concatenation of the lists. In some case it can be determined whether or not an arbitrary collection of blocks F will be a di%erence family, by the following procedure. Let B be a subset of an abelian group G. Then deHne the list of di&erence from B to be the list MB = (a − b: a; b ∈ B; a = b). When F = {Bi : i ∈ I } is a family of k-subsets of G, we deHne MF = i∈I MBi . If MF contains every nonzero element of G exactly times, then F is a (G; k; )-DF. Note that we here only consider those di%erence families without short blocks. We Hrst consider the constructions of complete sets of disjoint di%erence families in G(q), the additive group of GF(q). For convenience, we select and Hx, for each prime power q, a primitive element ! of GF(q). When e|(q − 1), we deHne the cosets e modulo the eth power, H0e = H e ; H1e ; : : : ; He−1 , by Hme = {!t : t ≡ m (mod e)}. We read e e : H e is the multhe subscripts modulo e, so that if a ∈ Hm and b ∈ Hne , then a · b ∈ Hm+n e tiplicative subgroup of GF(q) − {0} of order (q − 1)=e and index e. {H e ; H1e ; : : : ; He−1 } are the cyclotomic classes of GF(q) of index e. They evidently partition GF(q) − {0}. e The cyclotomic classes {H e ; H1e ; : : : ; He−1 } will be denoted by He . Given a set of e distinct elements in GF(q), if they belong to e di%erent cyclotomic e classes H e ; H1e ; : : : ; He−1 , then we say that this set of e elements forms a system of e }, and is denoted distinct representatives for the cyclotomic classes {H e ; H1e ; : : : ; He−1 e by SDRC(H ). Note that if q is even, then −1 = 1 is always an eth power in GF(q). If q is odd, then −1 ∈ H e if and only if 2e|(q − 1). In fact, −1 = !(q−1)=2 is an eth power if and only if (q − 1)=2 ≡ 0 (mod e). It will be convenient to introduce a multiplication of lists as follows: (ai : i ∈ I ) · (bj : j ∈ J ) = (ai · bj : i ∈ I; j ∈ J ):
Lemma 3.2. If there exists a (G(q); k; )-CDDF; then there exists a (G(q n ); k; )CDDF for every n ∈ N. Proof. Let {F0 ; : : : ; Fs−1 } be a (G(q); k; )-CDDF; where F‘ = {Bi‘ : i ∈ I (‘)}; 06 ‘ 6 s − 1; s = (k − 1)=; are the component (G(q); k; )-DDFs; such that MF‘ = ‘ ‘ ‘ i∈I (‘) MBi = (GF(q) − {0}) and Bi ∩ Bj = ∅ for all i; j ∈ I (l) where i = j; and for all ‘; 0 6 ‘ 6 s − 1; and also 06l6s−1 i∈I (l) Bil = GF(q) − {0}. Consider GF(q) as a subHeld of GF(q n ); then GF(q) − {0} is the subgroup H e of the eth powers in GF(q n ) where e = (q n − 1)=(q − 1). Now let R be any system of distinct representatives for the cosets He modulo H e in GF(q n ). Then R is a set of Held elements such that R·H e =GF(q n )−{0}. Consider the families F‘∗ ={rBi‘ : i ∈ I (‘); r ∈ R}; 0 6 ‘ 6 s−1. All elements of Bi‘ ; 0 6 ‘ 6 s−1; i ∈ I (‘); are in H e ; thus the blocks of F‘∗ are mutually disjoint since distinct elements of R belong to di%erent cosets of He and Bil ’s are mutually disjoint. Noting that the list of di%erences arising from the block rBi‘ is r·MBi‘ ;
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we have that MF‘∗ = r∈R i∈I (‘) r · MBi‘ = R · MF‘ = R · H e = (GF(q n ) − {0}). Thus every F‘∗ ; 0 6 ‘ 6 s − 1; is a (G(q n ); k; )-DDF. Meanwhile 06‘6s−1 B∈F∗ B = ‘ n ‘ e 06‘6s−1 i∈I (‘) r∈R rBi = R · (GF(q) − {0}) = R · H = GF(q ) − {0}; that is; the base ∗ blocks of F‘∗ ; 0 6 ‘ 6 s − 1; form a partition of GF(q n ) − {0}. Hence {F0∗ ; : : : ; Fs−1 } form a (G(q n ); k; )-CDDF. We note that Lemma 3.2 can also be deduced from Theorem 2.3 and Lemma 2.1. The next assertion can be checked easily, so we omit its proof. Theorem 3.3. Let q = 6t + 1 be a prime power. Then {F0 ; F1 }; where Fj = {!i+jt H 2t : 0 6 i 6 t − 1}; j = 0; 1; forms a (G(q); 3; 1)-CDDF. It can also be easily seen that the necessary condition for the existence of such a (G(q); 3; 1)-CDDF is q = 6t + 1 for some t ∈ N. By applying Theorem 2.3 with Lemma 3.2 and Theorem 3.3, we get the following result. Corollary 3.4. There exists a (3u; 3; 1)-CDDF; where the prime power factorization of u has the form u = uini where ui = 25; 55; or a prime congruent to 1 module 6. Now we use a result of Buratti (1995a) to establish a necessary and suOcient condition for the existence of a special (G(q); 5; 1)-CDDF. Lemma 3.5. (Buratti; 1995a). Let q = 20t + 1 be a prime power; 2e the highest power of 2 in t; and ' a primitive 5th root of unity in the ;eld GF(q). Then the necessary and su
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Next we consider the suOciency. Assume that the condition, which is described in Lemma 3.5, holds. Then, there is an integer 0 6 f 6 e such that ' + 1 is a 2f th power but not a 2f+1 th power in GF(q). Consider the following four subsets: X0 = {!2
f+1
X1 = {!2
i+j
f+1
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f };
i+2f +j
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f };
X2 = {!2t+2
f+1
i+j
X3 = {!2t+2
f+1
i+2f +j
: 0 6 i ¡ t=2f ; 0 6 j ¡ 2f }; : 0 6 i ¡ t=2f ; 0 6 j ¡ 2f }:
Buratti (1995a) proved that the family F0 = {xH 4t : x ∈ X0 } is a (G(q); 5; 1)-DF. Evidently, the base blocks of F0 are mutually disjoint. Similarly, prove that we can Fi = {xH 4t : x ∈ Xi }; i = 1; 2; 3, are also (G(q); 5; 1)-DDFs. Since 06i63 B∈Fi B form a partition of GF(q) − {0}, we know that {Fi : i = 0; 1; 2; 3}, where Fi = {xH 4t : x ∈ Xi }, forms a (G(q); 5; 1)-CDDF. Buratti (1995a) veriHed that the primes q = 20t + 1 ¡ 104 satisfying the condition mentioned above are the following: 41; 61; 241; 281; 401; 421; 601; 641; 661; 701; 761; 821; 881; 1361; 2441; 2801; 3001; 3121; 3881; 4001; 4201; 4561; 5281; 5441; 6481; 6521; 6761; 6841; 6961; 7841; 8161; 8641; 9001; 9521: We continue to investigate the case for odd block size ¿ 5. Theorem 3.7. Let q ≡ 1 (mod k(k − 1)) be a prime power; k = 2m + 1; ' a primitive kth root of unity in GF(q); and assume that {' − 1; : : : ; 'm − 1} is a system of distinct representatives for the cyclotomic classes Hm . Then there exists a (G(q); k; 1)-CDDF. Proof. Assume q − 1 = tk(k − 1) = 2tm(2m + 1). Let B = {1; '; : : : ; 'k−1 }. Then ' = !2tm . Put F0 = {B · !im : i = 0; : : : ; t − 1}; F1 = {B · !im+1 : i = 0; : : : ; t − 1}; .. . Fm−1 = {B · !im+m−1 : i = 0; : : : ; t − 1}; Fm = {B · !tm+im : i = 0; : : : ; t − 1}; Fm+1 = {B · !tm+im+1 : i = 0; : : : ; t − 1}; .. . F2m−1 = {B · !tm+im+m−1 : i = 0; : : : ; t − 1}:
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Wilson (1972) proved that if {' − 1; : : : ; 'm − 1} is an SDRC(Hm ) then F0 is a (G(q); k; 1)-DF; whose base blocks are evidently mutually disjoint. Similarly; we can prove that F1 ; : : : ; F2m−1are also (G(q); k; 1)-DFs with mutually disjoint base blocks. It can be checked that 06j62m−1 B∈Fj B form a partition of GF(q) − {0}. Hence {F0 ; : : : ; F2m−1 } forms a (G(q); k; 1)-CDDF. Wilson (1972) used a computer to check the validity of the hypothesis of Theorem 3.7 for several values of k and q. He succeeded in Hnding such a primitive kth root of unity in GF(q) for q = 337; 421; 463; 883; 1723; 3067; 3319 when k = 7, for q = 73; 1153; 1873; 2017 when k = 9, and for q = 76231 when k = 15. Recently, Buratti (1995b) gave necessary and suOcient conditions for the validity of the hypothesis of Theorem 3.7 for k = 7 and also gave a suOcient condition for k ¿ 7. The interested reader is referred to Buratti (1995b) for details. The above constructions are only applicable to the case when the block size is odd. What can we say about the case when the block size is even? Obviously we can not use the same idea to produce complete sets of disjoint di%erence families with even block size and = 1 in Hnite Helds. Here, we present a direct construction for complete sets of disjoint di%erence families which is applicable irrespective of the parity of the block size. This method of construction is rather di%erent from those described above. Let B = {b0 ; b1 ; : : : ; bk−1 }, where bi ∈ GF(q); 0 6 i 6 k − 1, and q = k(k − 1)t + 1; t ∈ N. We deHne the following k − 1 families of blocks of size k: Fj = {B · !k(k−1)i+kj : 0 6 i 6 t − 1};
0 6 j 6 k − 2:
In order for {F0 ; : : : ; Fk−2 } to be a (G(q); k; 1)-CDDF we need some suOcient conditions on the k-dimensional vector (b0 ; b1 ; : : : ; bk−1 ). Theorem 3.8. Let q = k(k − 1)t + 1 be a prime power. If there exist k elements of GF(q); b0 ; b1 ; : : : ; bk−1 ; satisfying the following conditions; then there exists a (G(q); k; 1)-CDDF. (C1 ) The set .{b0 ; : : : ; bk−1 } of k(k − 1) di&erences among {b0 ; : : : ; bk−1 } forms an SDRC(Hk(k−1) ); (C2 ) (b0 ; : : : ; bk−1 ) · (!kj : 0 6 j 6 k − 2) forms an SDRC(Hk(k−1) ). Condition (C1 ) is required for the di%erences arising from the blocks in Fj ; 0 6 j 6 k − 2, so that every nonzero element of GF(q) can occur precisely once, while Condition (C2 ) is required for the blocks of 06j6k−2 Fj to be a partition of GF(q)−{0}. Immediately we can know that t must be odd. For otherwise (b0 −b1 )=(b1 −b0 ) ∈ H k(k−1) , contradicting (C1 ). We focus our attention on the case when k = 4. Let (b0 ; b1 ; b2 ; b3 ) = (1; x; x2 ; x3 ) for some x ∈ GF(p) − {0; 1}, where p = 12t + 1 is a prime number and t is odd. If there exists an element x ∈ H112 such that 1 + x ∈ H312 ;
1 + x + x2 ∈ H512 ; or
1 + x ∈ H912 ;
1 + x + x2 ∈ H512 ; or
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Table 1 p
!
(b0 ; b1 ; b2 ; b3 )
61 109 181 397
2 6 2 5
(1; 8; 27; 24) (1; 6; 100; 33) (1; 2; 168; 151) (1; 5; 324; 343)
1 + x ∈ H412 ;
1 + x + x2 ∈ H912 ; or
12 1 + x ∈ H10 ;
1 + x + x2 ∈ H912 ;
then it can be easily checked that (1; x; x2 ; x3 ) satisHes our Conditions (C1 ) and (C2 ). By using Weil’s theorem on character sums, and by the aid of a computer, Lam and Miao (1999) found such an element x of GF(p) satisfying one of the above systems for every prime p = 12t + 1; t odd, except for p = 13; 61; 109; 181 and 397. For the remaining Hve cases, we ran a computer program to Hnd (b0 ; b1 ; b2 ; b3 ) which satisfy our Conditions (C1 ) and (C2 ). We found solutions for all these remaining cases except for p=13. We list p; ! and (b0 ; b1 ; b2 ; b3 ) in Table 1, where ! is the primitive element of GF(p) we used in our computation. We did not Hnd any suitable (b0 ; b1 ; b2 ; b3 ) for p = 13 by exhaustive search. In fact, the following 4-subsets are all the possible di%erence families with k = 4 and = 1 over G(13). {0; 1; 3; 9}; {0; 1; 4; 6}; {0; 1; 5; 11}; {0; 1; 8; 10}; {0; 2; 3; 7}; {0; 2; 5; 6}; {0; 2; 8; 12}; {0; 2; 9; 10}; {0; 3; 4; 11}; {0; 3; 5; 12}; {0; 4; 5; 7}; {0; 4; 10; 12}; {0; 6; 8; 9}; {0; 6; 10; 11}; {0; 7; 8; 11}; {0; 7; 9; 12}; {1; 2; 4; 10}; {1; 2; 5; 7}; {1; 2; 6; 12}; {1; 2; 9; 11}; {1; 3; 4; 8}; {1; 3; 6; 7}; {1; 3; 10; 11}; {1; 4; 5; 12}; {1; 5; 6; 8}; {1; 7; 9; 10}; {1; 7; 11; 12}; {1; 8; 9; 12}; {2; 3; 5; 11}; {2; 3; 6; 8}; {2; 3; 10; 12}; {2; 4; 5; 9}; {2; 4; 7; 8}; {2; 4; 11; 12}; {2; 6; 7; 9}; {2; 8; 10; 11}; {3; 4; 6; 12}; {3; 4; 7; 9}; {3; 5; 6; 10}; {3; 5; 8; 9}; {3; 7; 8; 10}; {3; 9; 11; 12}; {4; 5; 8; 10}; {4; 6; 7; 11}; {4; 6; 9; 10}; {4; 8; 9; 11}; {5; 6; 9; 11}; {5; 7; 8; 12}; {5; 7; 10; 11}; {5; 9; 10; 12}; {6; 7; 10; 12}; {6; 8; 11; 12}:
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It can be easily checked that no three of them can be mutually disjoint. So we can assert that there is no (G(13); 4; 1)-CDDF. Furthermore, since there is no other group of order 13, we can even say that there is no complete set of disjoint di%erence families of block size k = 4 and index = 1 over any group of order 13. Summarizing the above, we can get the following result. Theorem 3.9. There exists a (G(p); 4; 1)-CDDF whenever p = 12t + 1 is a prime number and t is odd; except for p = 13. We do not know, by now, whether there is a similar result for every even block size k ¿ 6. Weil’s theorem could provide a bound p0 (k) for each even block size k ¿ 6 such that for any prime p = k(k − 1)t + 1, where t is odd and p ¿ p0 (k), there exists a (G(q); k; 1)-CDDF. A computer could also be used to Hnd a proper element x ∈ GF(q) for each small prime p = k(k − 1)t + 1; t odd, to guarantee the existence of a (G(q); k; 1)-CDDF for even k ¿ 6. But it seems impractical for us at this moment to ask a computer to Hnd such an element x ∈ GF(q) for all prime p ¡ p0 (k) with p = k(k − 1)t + 1 and t is odd for any even block size k ¿ 6. A new method is desired for a complete solution to this problem. Sometimes, the properties of Hnite rings can also be used to produce various combinatorial structures. For example, complete sets of disjoint di%erence families can also be constructed directly from Hnite rings. Let R be a commutative ring of odd order admitting a unit ' of order k, where k is an odd integer, such that {' − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Consider the relation ∼ deHned in R − {0} by x∼y
if and only if
y = 's x for some s ∈ {0; 1; : : : ; k − 1}:
It is easy to see that ∼ is an equivalence relation whose equivalence class is the set r' = {r; r'; : : : ; r'k−1 }; r ∈ R − {0}. Since {' − 1; : : : ; '(k−1)=2 − 1} is a set of units, each of these sets has actually size k. Hence a system of distinct representatives for the equivalence classes of ∼ has cardinality (|R| − 1)=k. Let X be such a system. If there exists a set X0 such that X =±X0 , then {X0 '; −X0 '} forms a (G; k; (k −1)=2)-CDDF, where G is the additive group of R. This can be checked as follows. The di%erences arising from '; M', is ±' · (' − 1; : : : ; '(k−1)=2 − 1). Then M(X0 ') = X0 · M' = X0 · ±'·('−1; : : : ; '(k−1)=2 −1)=±X0 ·'·('−1; : : : ; '(k−1)=2 −1)=X ·'·('−1; : : : ; '(k−1)=2 −1) = (G − {0}) · (' − 1; : : : ; '(k−1)=2 − 1) = ((k − 1)=2)(G − {0}); M(−X0 ') = −MX0 · ' = ((k − 1)=2)(G − {0}), and (X0 · ') ∪ (−X0 · ') = ±X0 · ' = X · ' = G − {0}. Lemma 3.10. Let R be a commutative ring of odd order admitting a unit of order k; where k is an odd integer; such that {' − 1; : : : ; 'k−1 − 1} is a set of units of R. Let X be a system of distinct representatives for the equivalence classes of ∼ de;ned above. Then; there exists a set X0 such that X = ±X0 . Proof. For any x ∈ X we claim that x' ∩ (−x') = ∅. Otherwise there would exist x0 ∈ X and s ∈ {0; 1; : : : ; k −1} such that −x0 =x0 's . Since ' is a unit of order k; −x0 'k = x0 's ; which means −x0 =x0 'k−s . So we know that x0 'k−s =x0 's ; which implies x0 ('k−2s −1) = 0. If k − 2s ≡ 0 (mod k); then by our hypothesis; 'k−2s − 1 is a unit; and then x0 = 0;
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which is impossible. So we must have that k − 2s ≡ 0 (mod k); i.e. 2s ≡ 0 (mod k). Since k is odd; s ≡ 0 (mod k); and thus s=0. This means −x0 =x0 ; which is impossible too since |R| is odd. Therefore, we can separate X into two sets X0 and −X0 so that x ∈ X0 if and only if −x ∈ (−X0 ). Here we note that the additive group G of the ring R contains no nonzero element which is its own additive inverse. As a summary, we have the following result. Theorem 3.11. Let R be a commutative ring of odd order admitting a unit ' of order k; where k is an odd integer; such that {' − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Then there exists a (G; k; (k − 1)=2)-CDDF; where G is the additive group of R. For example, we have the following two consequences. Corollary 3.12. Let v and k be odd integers such that p ≡ 1 (mod k) holds for each prime p in v. Then there exists a (Zv ; k; (k − 1)=2)-CDDF. Proof. Let v = i∈I pini be the prime power factorization of v. Then by the Chinese Remainder Theorem; we can identify Zv with the ring R = ⊕i∈I Zpini . Buratti (1997) described a unit ' of order k in R such that {' − 1; '2 − 1; : : : ; '(k−1)=2 − 1} is a set of units of R. Then the assertion follows from Theorem 3.11. Corollary 3.13. Let v and k be odd integers and v = i∈I pini the prime power factorization of v such that pini ≡ 1 (mod k) for each i ∈ I . Then there exists a (G(v); k; (k − 1)=2)-CDDF; where G(v) is the additive group of the Galois ring GR(v) of order v. Proof. Buratti (1997) also described a unit ' of order k in GR(v) such that {' − 1; '2 − 1; : : : ; '(k−1)=2 − 1} is a set of units of GR(v). Then the assertion follows from Theorem 3.11. In order to clarify the above results, let us see the following example. Consider the Galois ring Z91 . It admits a unit 9 of order 3 such that 9 − 1 = 8 is still a unit. Then the two di%erence families described below form a (Z91 ; 3; 1)-CDDF. {{{1; 9; 81}; {2; 18; 71}; {3; 27; 61}; {4; 36; 51}; {5; 45; 41}; {6; 54; 31}; {7; 63; 21}; {8; 72; 11}; {12; 17; 62}; {13; 26; 52} {14; 35; 42}; {15; 44; 32}; {16; 53; 22}; {23; 25; 43}; {24; 34; 33}} ; {{10; 90; 82}; {20; 89; 73}; {30; 88; 64}; {40; 87; 55}; {46; 50; 86}; {37; 60; 85}; {28; 70; 84}; {19; 80; 83};
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{29; 79; 74} {39; 78; 65}; {49; 77; 56}; {47; 59; 76}; {38; 69; 75}; {48; 68; 66} {67; 57; 58}}} : 4. An application to one-factorizations of complete graphs In the remainder of this paper, we mention two applications of complete sets of disjoint di%erence families. First, we describe an application to the construction of a special kind of one-factorizations of complete graphs. Recall that a complete graph Kn is a graph on n vertices such that every pair of vertices are adjacent, that is, every pair of vertices is connected by an edge. A one-factor in a complete graph Kn is a set of edges in which every vertex appears precisely once. In other words, it is a regular spanning subgraph of degree 1. A one-factorization of Kn is a way of partitioning the edge set of Kn into one-factors. Let G be an abelian group of order 2m + 1, and S be a (G; 2; 1)-CDDF. Then S is in fact a family of unordered pairs {{xi ; yi }: 1 6 i 6 m} which satisHes the following two properties: (P1) {xi : 1 6 i 6 m} ∪ {yi : 1 6 i 6 m} = G − {0}, (P2) {±(xi − yi ): 1 6 i 6 m} = G − {0}. Historically, such family of unordered pairs is called a starter in G, which was introduced by Stanton and Mullin (1968) for the direct construction of Room squares. Later on, starters were also proved to be useful in the construction of one-factorizations of complete graphs, as the following shows. Theorem 4.1. If there is a starter S in an abelian group G of order 2m + 1; then the 2m + 1 factors {∞; x} ∪ {S + x}; where x ranges through G; form a one-factorization of K2(m+1) based on {∞} ∪ G. This construction can be found, for example in Wallis (1992). Here we describe another construction of a one-factorization of K2(2m+1) , which is also generated by a starter. Let S be a starter in an abelian group G of order 2m + 1. Then |S| = m. Let the complete graph K2(2m+1) be based on Z2 ⊕G. Then it can be checked that the edge set developed modulo (Z2 ; G) from {{(0; xi ); (0; yi )}: {xi ; yi } ∈ S}∪{{(0; xi ); (1; yi )}: {xi ; yi } ∈ S} and modulo (−; G) from {(0; 0); (1; 0)} is the edge set of this K2(2m+1) , while FI (S) = {{(0; xi ); (0; yi )}: {xi ; yi } ∈ S} ∪ {{(1; xi ); (1; yi )}: {xi ; yi } ∈ S} ∪ {(0; 0); (1; 0)} is a one-factor. Develop it modulo (−; G) we get 2m + 1 one-factors FI (S) + (−; g), where g ∈ G. Also for each pair {xi ; yi } ∈ S; 1 6 i 6 m; FII (S; i) = {{(0; xi ); (1; yi )} mod(−; G): {xi ; yi } ∈ S} is a one-factor. Develop it modulo (Z2 ; −) we get 2 one-factors FII (S; i) + (z; −), where z ∈ Z2 . Altogether we get 4m + 1 one-factors, which form a one-factorization of the K2(2m+1) based on Z2 ⊕ G. Obviously this one-factorization is invariant under the action of Z2 ⊕ G, with one factor orbit of length 2m + 1, and m factor orbits of length 2.
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Theorem 4.2. If there is a starter in an abelian group G of order 2m + 1; then there is a one-factorization of the K2(2m+1) based on Z2 ⊕G which is invariant under Z2 ⊕G. Note that Theorem 4.2 was Hrst proved for the case when G is a cyclic group of order 2m + 1, that is, G = Z2m+1 , by Hartman and Rosa (1985). We investigate further properties on the intersections of such kind of onefactorizations. Let S = {{si ; ti }: 1 6 i 6 m} and T = {{ui ; vi }: 1 6 i 6 m} be two starters in G. Without loss of generality, we may assume that si − ti = ui − vi for all i; 1 6 i 6 m. If ui −si =uj −sj can imply i =j, and if ui = si for all i; 1 6 i 6 m, then S and T are said to be orthogonal. Two orthogonal starters S and T can produce two one-factorizations of the K2(2m+1) based on Z2 ⊕G which are invariant under Z2 ⊕G. Let hi =si −ui , for all i; 1 6 i 6 m. Then, by the deHnition of orthogonal starters, we can get the following result on the intersection problem of such two one-factorizations. (FI (S) + (−; g)) ∩ (FI (T ) + (−; g) + (−; hi )) = {{(0; si ); (0; ti )}; {(1; si ); (1; ti )}} for all g ∈ G and for all hi ; 1 6 i 6 m; (FI (S) + (−; g)) ∩ (FI (T ) + (−; g)) = {{(0; 0); (1; 0)}}
for all g ∈ G;
(FI (S) + (−; g)) ∩ (FI (T ) + (−; g) + (−; h)) =∅
for all g ∈ G and for all h ∈ G − {0} ∪ {hi : 1 6 i 6 m};
(FI (S) + (−; g)) ∩ (FII (T; i) + (z; −)) =∅
for all g ∈ G; all z ∈ Z2 ; and all i; 1 6 i 6 m;
(FI (T ) + (−; g)) ∩ (FII (S; i) + (z; −)) =∅
for all g ∈ G; all z ∈ Z2 ; and all i; 1 6 i 6 m;
(FII (S; i) + (z; −)) ∩ (FII (T; i) + (z; −)) = FII (S; i) + (z; −) = FII (T; i) + (z; −)
for all z ∈ Z2 ;
(FII (S; i) + (z; −)) ∩ (FII (T; i) + (z + 1; −)) =∅
for all z ∈ Z2 and all i; 1 6 i 6 m;
(FII (S; i) + (z1 ; −)) ∩ (FII (T; j) + (z2 ; −)) =∅
for all i = j; 1 6 i; j 6 m; all z1 ∈ Z2 and all z2 ∈ Z2 :
The interested reader is referred to Lindner and Wallis (1982) for a detailed discussion on the intersection problem of two one-factorizations.
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5. An application to cyclically resolvable cyclic Steiner systems Let V be a set of v elements, and B a collection of k-subsets of V called blocks. A balanced incomplete block design (BIBD), denoted by (v; k; )-BIBD, is a pair (V; B) such that every pair of distinct elements of V is contained in precisely blocks of B. A BIBD with = 1 is called a Steiner 2-system, while a Steiner 2-system with block size k = 3 is a Steiner triple system. An automorphism of a BIBD (V; B) is a permutation on V whose induced mapping from B to B is a permutation too. The set of all automorphisms of a BIBD forms a group, the full automorphism group. Any of its subgroups is called an automorphism group of the BIBD. A (v; k; )-BIBD admitting a cyclic and element-regular automorphism group is called a cyclic (v; k; )-BIBD. For a cyclic BIBD (V; B), the set V of elements can be identiHed with Zv . In this case, the BIBD has an automorphism : i → i + 1 (mod v). If a block orbit is of length v, then this block orbit is said to be full, otherwise short. The block orbit which contains the following block is called a regular short block orbit (v=k)Zk = {0; v=k; : : : ; (k − 1)v=k}: A resolution of a BIBD (V; B) is a partition of B into resolution classes each of which is a partition of V. If such a resolution exists, then (V; B) is said to be resolvable. Let G and R be an automorphism group and a resolution of a BIBD (V; B), respectively. If R is Hxed by G, i.e. if Rig ∈ R for all (g; Ri ) ∈ G × R, we say that R is G-invariant. In this case, we also say that (V; B) is G-invariantly resolvable. Trivially, “{1v }-invariantly resolvable” simply means “resolvable”, while a Zv invariantly resolvable cyclic (v; k; )-BIBD is commonly called cyclically resolvable cyclic (v; k; )-BIBD, and denoted by (v; k; )-CRCB. It is clear now that the one-factorization of K2(2m+1) we constructed in Section 4 is in fact a Z2 ⊕ G-invariantly resolvable (2(2m + 1); 2; 1)-BIBD. In this section, we will consider a special case of such BIBDs with k = 3 and = 1, while G is a cyclic group, i.e. cyclically resolvable cyclic Steiner triple systems. Of course, we can adapt it for non-cyclic cases. Cyclically resolvable cyclic Steiner 2-systems and related structures have been investigated by several authors, see for example (Genma et al., 1997; Kageyama and Miao, 1998; Lam and Miao, 1999; Mishima and Jimbo, 1997). Mishima and Jimbo (1997) classiHed cyclically resolvable cyclic Steiner 2-systems into three types (T1), (T2) and (T3), according to their relation with cyclic quasiframes, cyclic semiframes, or cyclically resolvable cyclic group divisible designs. In particular, in type (T2), each block in the regular short block orbit is in a distinct resolution class. The reader is referred to Mishima and Jimbo (1997) for their formal deHnitions. Before we describe our method of construction, we need some necessary condition on the number of elements in a cyclically resolvable cyclic Steiner triple system of type (T2). An obvious necessary condition for the existence of a resolvable cyclic Steiner triple system with number of elements v is v ≡ 3 (mod 6). In the case of a cyclically resolvable cyclic Steiner triple system of type (T2), a stronger condition is needed.
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Lemma 5.1. (Genma et al.; 1997). For a (3u; 3; 1)-CRCB of type (T 2); u ≡ 1 (mod 6) holds. So we can assume that u=6t+1 for some integer t ∈ N∪{0}. Since gcd(3; u)=1, we know that Z3u Z3 ⊕Zu . In this section, instead of using Z3u as the set of elements and : i → i + 1 (mod 3u) as the automorphism, we will use Z3 ⊕ Zu as the set of elements and : (i; j) → (i; j) + (1; 1) (mod (3; u)) as the automorphism. With this representation, it is easier to see the structure of the CRCB. Now we provide our direct construction for (3u; 3; 1)-CRCB of type (T2) based on the set of elements V = Z3 ⊕ Zu , where u = 6t + 1 for some t ∈ N ∪ {0}. Theorem 5.2. If there exists a (Z6t+1 ; 3; 1)-CDDF then there exists a (3(6t + 1); 3; 1)CRCB of type (T 2). Proof. We Hrst construct the following 3t + 1 blocks: A = {(0; 0); (1; 0); (2; 0)}; Bi = {(0; bi0 ); (0; bi1 ); (0; bi2 )};
1 6 i 6 t;
Ci = {(0; ci0 ); (1; ci1 ); (2; ci2 )};
1 6 i 6 t;
Di = {(0; di0 ); (1; di1 ); (2; di2 )};
1 6 i 6 t:
We try to Hnd some suOcient conditions on bij ; cij ; dij ; 1 6 i 6 t; 0 6 j 6 2; so that we could get a (3u; 3; 1)-CRCB of type (T2); where V = Z3 ⊕ Z6t+1 ; B = {A (mod (−; 6t + 1))} ∪ {Bi (mod (3; 6t + 1)): 1 6 i 6 t} ∪ {Ci (mod (3; 6t + 1)): 1 6 i 6 t} ∪ {Di (mod (3; 6t + 1)): 1 6 i 6 t}; and a base resolution class of a resolution class orbit modulo (−; 6t + 1) is {A} ∪ {Bi (mod (3; −)): 1 6 i 6 t} ∪ {Ci (mod (3; −)): 1 6 i 6 t}; while a base resolution class of a resolution class orbit modulo (3; −) is {Di (mod (−; 6t + 1)): 1 6 i 6 t}: The conditions are described in the following: (C1) {±(bi0 − bi1 ); ±(bi0 − bi2 ); ±(bi1 − bi2 ): 1 6 i 6 t} = Z6t+1 − {0}; (C2) {ci0 − ci1 ; ci1 − ci2 ; ci2 − ci0 ; di0 − di1 ; di1 − di2 ; di2 − di0 } = Z6t+1 − {0}; (C3) {bij ; cij : 1 6 i 6 t; 0 6 j 6 2} = Z6t+1 − {0}.
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Condition (C1) is required for the pure di%erences arising from the blocks in B; Condition (C2) is required for the mixed di%erences arising from the blocks in B; and Condition (C3) is required for the resolution class which contains a block from the regular short block orbit A = {A (mod (−; 6t + 1))}. If we take di0 = ci0 ; di1 = ci2 and di2 = ci1 , then {ci0 − ci1 ; ci1 − ci2 ; ci2 − ci0 ; di0 − di1 ; di1 − di2 ; di2 − di0 } = {±(ci0 − ci1 ); ±(ci1 − ci2 ); ±(ci0 − ci2 )}. That is, Condition (C2) becomes Condition (C2 ) (C2 ) {±(ci0 − ci1 ); ±(ci1 − ci2 ); ±(ci0 − ci2 )} = Z6t+1 − {0}: Therefore, what we need is a doubly disjoint di%erence family (Z6t+1 ; 3; 1)-CDDF, where F0 = {{bi0 ; bi1 ; bi2 }: 1 6 i 6 t} and F1 = {{ci0 ; ci1 ; ci2 }: 1 6 i 6 t}. It completes the proof. As an immediate consequence, we have the following result. Corollary 5.3. There exists a (3u; 3; 1)-CRCB of type (T 2); where the prime power factorization of u has the form u = uini where ui = 25; 55; or a prime congruent to 1 modulo 6. Proof. Such a (u; 3; 1)-CDDF can be found in Corollary 3.4. Acknowledgements The authors would like to thank the referees for their valuable suggestions in revising this paper. References Abel, R.J.R., 1996. Di%erence families. In: Colbourn, C.J., Dinitz, J.H. (Eds.), The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 270–287. Buratti, M., 1995a. Improving two theorems of Bose on di%erence families. J. Combin. Des. 3, 15–24. Buratti, M., 1995b. On simple radical di%erence families. J. Combin. Des. 3, 161–168. Buratti, M., 1997. From a (G; k; 1) to a (Ck ⊕ G; k; 1) di%erence family. Des. Codes Cryptogr. 11, 5–9. Colbourn, M.J., Colbourn, C.J., 1984. Recursive constructions for cyclic block designs. J. Statist. Plann. Inference 10, 97–103. Colbourn, C.J., de Launey, W., 1996. Di%erence matrices. In: Colbourn, C.J., Dinitz, J.H. (Eds.), CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 287–297. Dinitz, J.H., 1996. Starters. In: Colbourn, C.J., Dinitz, J.H. (Eds.), CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, pp. 467–473. Dinitz, J.H., Rodney, P., 1997. Disjoint di%erence families with block size 3. Utilitas Math. 52, 153–160. Dinitz, J., Shalaby, N., Block disjoint di%erence families for Steiner triple systems: v ≡ 3 mod 6. J. Statist. Plann. Inference, Special issue dedicated to the memory of Professor Sumiyashu Yamamoto. Furino, S., Miao, Y., Yin, J., 1996. Frames and Resolvable Designs: Uses, Constructions, and Existence. CRC Press, Boca Raton, FL. Genma, M., Mishima, M., Jimbo, M., 1997. Cyclic resolvability of cyclic Steiner 2-designs. J. Combin. Des. 5, 177–187.
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