- Email: [email protected]
Metamorphosis of simple twofold triple systems into twofold (K4−e) -designs
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 40 (2013) 217–222 www.elsevier.com/locate/endm
Metamorphosis of simple twofold triple systems into twofold (K4 − e)-designs Yanxun Chang, Tao Feng 1,2 Institute of Mathematics Beijing Jiaotong University Beijing, P. R. China
Giovanni Lo Faro, Antoinette Tripodi 3,4 Department of Mathematics and Computer Science University of Messina Messina, Italy
Abstract In this note, we show that there exists a metamorphosis of a simple twofold triple system of order v into a twofold (K4 − e)-design of order v if and only if v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 10. Keywords: twofold triple system, twofold (K4 − e)-design, metamorphosis
1 2 3 4
Email: [email protected] (Y. Chang), [email protected] (T. Feng). Supported by NSFC Grant 61071221 (Y. Chang) and NSFC Grant 10901016 (T. Feng). Email: [email protected] (G. Lo Faro), [email protected] (A. Tripodi). Supported by P.R.A. and I.N.D.A.M.(G.N.S.A.G.A.) (G. Lo Faro).
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.039
218
1
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
Introduction
Let λ be a positive integer, H be a simple finite graph, and let λH denote the graph H with each of its edges replicated λ times. Let G be a simple finite graphs. A (λH, G)-design is a pair (X, B), where X is the vertex set of H and B is a collection of subgraphs (called blocks) of λH, such that each block is isomorphic to G, and each edge of λH is contained in exactly λ blocks of B. A (λH, G)-design (X, B) is called simple if B contains no repeated blocks. In what follows, the graph Kv denotes the complete graph with v vertices; K4 − e denotes a graph obtained from K4 by deleting one edge; [a, b, c − d] denotes the copy of K4 − e with vertices a, b, c, d and the edge cd removed. A (λKv , G)-design is called a λ-fold (K4 −e)-design of order v. A (λKv , Kk )design is called a balanced incomplete block design and denoted by (v, k, λ)BIBD. A (v, 3, 2)-BIBD is called a twofold triple system of order v. Let (X, B) be a simple twofold triple system of order v. For every x, y ∈ X, x = y, the pair {x, y} is contained in exactly two different triples, say, {x, y, z} and {x, y, w}. Any two blocks B1 , B2 ∈ B such that |B1 ∩ B2 | = 2 form a matched pair. Suppose now that there is a partition of B into |B|/2 matched pairs. If we replace the double edge {x, y} with its corresponding single edge {x, y} from a matched pair {x, y, z}, {x, y, w}, we are left with the (K4 − e) [x, y, z − w]. If C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and if the collection of deleted edges can be reassembled into a collection D of (K4 − e)s, then (X, C ∪ D) is a twofold (K4 − e)-design of order v, which is said to be a metamorphosis of the simple twofold triple system (X, B) and denoted by an M(2K3 > K4 − e, v) or more briefly by an M(v). For this metamorphosis M(v) to exist, it is necessary that both the simple twofold triple system and the twofold (K4 − e)-design of order v exist, that is, one must have v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 6. Example 1.1 We here give an example of an M(10). Take X = {0, 1, . . . , 9}. We construct a metamorphosis of a simple twofold triple system of order 10 (X, B) into a twofold (K4 − e)-design of order 10 (X, C ∪ D) as follows: C = {[0, 1, 4 − 5], [0, 2, 6 − 7], [0, 3, 4 − 9], [1, 2, 6 − 7], [1, 3, 5 − 9], [4, 5, 7 − 8], [4, 2, 8−9], [4, 6, 9−1], [5, 2, 9−3], [5, 6, 8−0], [7, 8, 0−1], [7, 3, 4−6], [7, 9, 5−6], [8, 3, 2 − 6], [8, 9, 0 − 1]} and D = {[0, 1, 2 − 3], [4, 5, 2 − 6], [7, 8, 3 − 9]}, where C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and the collection of deleted edges can be reassembled into a collection D of (K4 − e)s. Note that it is not necessary to list B explicitly since the blocks of B can be deduced from
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
219
C. For instance, [0, 1, 4 − 5] of C contributes two blocks {0, 1, 4} and {0, 1, 5} of B. Other results on problems in Design Theory and, in particular, on metamorphoses can be found in [2, 3, 6–13]; for more information, the interested reader may refer to [4] and the references therein. As the main result, we are to prove the following theorem. Theorem 1.2 There exists an M(v) if and only if v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 10.
2
Preliminaries
Let H = {H1 , H2 , . . . , Hm } be a partition of a finite set X into subsets (called holes), where |Hi | = ni for 1 ≤ i ≤ m. Let Kn1 ,n2 ,...,nm be the complete multipartite graph on X with the i-th part on Hi , and G be a subgraph of Kn1 ,n2 ,...,nm . A λ-fold holey G-design is a triple (X, H, B) such that (X, B) is a (λKn1 ,n2 ,...,nm , G)-design. The hole type (or type) of the λ-fold holey Gdesign is the multiset {n1 , n2 , . . . , nm }. We use an “exponential” notation to describe hole types: the hole type g1u1 g2u2 · · · grur denotes ui occurrences of gi for 1 ≤ i ≤ r. When G is the complete graph Kk , a λ-fold holey Kk -design is said to be a (k, λ)-group divisible design (briefly (k, λ)-GDD). When G is K4 − e, a λ-fold holey G-design is called a λ-fold (K4 − e)-HD. Let (X, G, B) be a simple (3, 2)-GDD of group type {n1 , n2 , . . . , nm }. Suppose that |B| is even and there is a partition of B into |B|/2 matched pairs. If C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and if the collection of deleted edges can be reassembled into a collection D of (K4 − e)s, then (X, G, C ∪ D) is a twofold (K4 − e)-HD with type {n1 , n2 , . . . , nm } which is said to be a metamorphosis of the simple (3, 2)-GDD (X, G, B). We usually denote the metamorphosis by M({n1 , n2 , . . . , nm }), or M(g1u1 g2u2 · · · grur ) if there are ui occurrences of gi in the hole type for 1 ≤ i ≤ r. An M(1v ) is actually an M(v). The following constructions are variations of standard constructions for group divisible designs in combinatorial design theory. Construction 2.1 (Weighting Construction) Suppose that (X, G, A) is a KGDD, and let ω : X −→ Z + ∪ {0} be a weight function. For every block A ∈ A, suppose that there is an M({ω(x) : x ∈ A}). Then there exists an M({ x∈G ω(x) : G ∈ G}).
220
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
Construction 2.2 (Filling Construction) Let a be a nonnegative integer. Suppose that there exists an M({g1 , g2 , . . . , gs }). If there is an M(1gi a1 ) for 1 ≤ i ≤ s − 1, then there exists an M(1v−gs (gs + a)1 ), where v = si=1 gi . If further there is an M(gs + a), then there exists an M(v + a). In order to use Constructions 2.1 and 2.2, we can show the following results by computer search. Lemma 2.1 There exists an M(g n ) for (g, n) ∈ {(5, 3), (5, 4), (10, 3), (10, 4), (15, 3), (15, 4), (15, 5)}. Lemma 2.2 There exist an M(1n h1 ) for (n, h) ∈ {(5, 4), (15, 6), (26, 10)}.
3
Main theorem
Lemma 3.1 There is no M(6). Proof. Suppose that there were an M(6). Then there were a simple twofold triple system of order 6 (X, B) whose blocks could be partitioned into 5 matched pairs, and only one K4 − e were reassembled from double edges, say, [a, b, c − d]. It means that a should occur in 6 different blocks of B: {a, b, ∗}, {a, b, ∗}, {a, c, ∗}, {a, c, ∗}, {a, d, ∗}, {a, d, ∗}. However, the number of blocks containing every given point of X in a twofold triple system of order 6 is 5, a contradiction. 2 Lemma 3.2 There exists an M(v) for v ≡ 0, 1, 6 (mod 15) and v ≥ 15. Proof. For v ≡ 0, 1, 6 (mod 15) and v ≥ 15, v can be written as v = 15n + h where n ≥ 1 and h = 0, 1, 6. When (n, h) ∈ {(1, 0), (1, 1), (1, 6), (2, 1)}, we can find an M(v) by computer search. When (n, h) = (2, 0), apply Construction 2.2 with an M(103 ) and an M(10), which exists by Lemma 2.1 and Example 1.1, respectively, to obtain an M(30). When (n, h) = (2, 6), apply Construction 2.2 with an M(126 101 ) from Lemma 2.2 and an M(10) to obtain an M(36). When n ≥ 3 and n = 6, 8, there is a (n, {3, 4, 5}, 1)-PBD (see [1]), that is, a {3, 4, 5}-GDD of type 1n . Give every point of the GDD weight 15 and apply Construction 2.1 to get an M(15n ), where the required M(15t ) for t = 3, 4, 5 come from Lemma 2.1. Then applying Construction 2.2 with an M(115 h1 ) and an M(15 + h), we obtain an M(v), where the required M(115 61 ) is from Lemma 2.2. When (n, h) ∈ {(6, 0), (6, 1), (8, 0), (8, 1)}, there exists a 3-GDD of type n/2 2 (see [5]). Give every point of the 3-GDD weight 15 and apply Construction 2.1 with an M(153 ) to get an M(30n/2 ). Then by Construction 2.2 with an
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
221
M(30 + h), we have the required M(v). When (n, h) ∈ {(6, 6), (8, 6)}, a 3GDD of type 3n−2 71 exists (see [5]). Give each point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(15n−2 351 ), where the required M(53 ) comes from Lemma 2.1. By Construction 2.2 with an M(16) and an M(36), we have the required M(v). 2 Lemma 3.3 There exists an M(v) for v ≡ 10 (mod 15) and v ≥ 10. Proof. For v ≡ 10 (mod 15) and v ≥ 10, v can be written as v = 5(3n + 2) where n ≥ 0. When n = 0, the conclusion follows by Example 1.1. When n = 1, we can find an M(25) by computer search. When n ≥ 2 is even, a 3-GDD of type 2(3n+2)/2 exists (see [5]). Give every point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(10(3n+2)/2 ). By Construction 2.2 with an M(10), there exists an M(v). When n ≥ 5 is odd, there is a 3-GDD of type 3n−1 51 (see [5]). Give every point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(15n−1 251 ), where the required M(53 ) comes from Lemma 2.1. Apply Construction 2.2 with an M(15) and an M(25) to get an M(v). When n = 3, there exists a 4-GDD of type 34 (it is equivalent to two mutually orthogonal Latin squares of order 3), and deleting a point of the fourth group, we obtain a {3, 4}-GDD of type 33 21 . Give every point of the GDD weight 5 and apply Construction 2.1 to get an M(153 101 ), where the required M(53 ) and M(54 ) come from Lemma 2.1. Then by Construction 2.2 with an M(10) and an M(15), there exists an M(55). 2 Combing the results of Lemmas 3.1, 3.2 and 3.3, we complete the proof of Theorem 1.2.
References [1] Abel, R.J.R., F.E. Bennett, M. Grig, PBD-closures, CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds) CRC Press (2012), 247–255. [2] Amato, A., M. Gionfriddo, L. Milazzo, 2-regular equicolourings for P(4)-designs, Discrete Math. 312 (2012), 2252–2261. [3] Berardi, L., M.Gionfriddo , R. Rota, Balanced and strongly balanced P(k)designs, Discrete Math. 312 (2012), 633–636. [4] Billington, E. J., Metamorphoses of Graph Designs, URL: http://users.monash.edu.au/ accmcc/talks/TalkBillington.pdf.
222
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
[5] Colbourn, C. J., D. G. Hoffman, and R. Rees, A new class of group divisible designs with blocks size three, J. Combin. Theory (A), 59 (1992), 73–89. [6] Chang, Y., G. Lo Faro and A. Tripodi, The spectrum of M eta(K4 − e > K3 + e, λ) with any λ, Utilitas Math. 72 (2007), 3–22. [7] Chang, Y., and G. Lo Faro, Intersection numbers of Kirkman triple systems, J. Combin. Theory (A), 2 (1999), 348–361. [8] Chang, Y., and G. Lo Faro, The flower intersection problem for Kirkman triple systems, J. Stat. Plann. and Inf., 110 (1-2)(2003), 159–177. [9] Chang, Y., G. Lo Faro and G. Nordo, The fine structures of three latin squares, Journal of Combinatorial Designs, 14 (2)(2006), 859–110. [10] Chang, Y., G. Lo Faro and A. Tripodi, Determining the spectrum of M eta(K4 + e > K4 , λ) for any λ, Discrete Math., 308 (2008), 439–456. [11] Gionfriddo, M., S. Kucukcifci, L. Milazzo , Balanced and Strongly Balanced 4-Kite designs, Utilitas Mathematica 91 (2013). [12] Lindner, C.C., and A. Tripodi, The metamorphosis of K4 − e designs into maximum packings of Kn with 4-cycles, Ars Combinatoria, 75 (2005), 333–349. [13] Lo Faro, G., A. Tripodi, The Doyen-Wilson theorem for kite systems, Discrete Math., 306 (2006), 2695–2701.
Electronic Notes in Discrete Mathematics 40 (2013) 217–222 www.elsevier.com/locate/endm
Metamorphosis of simple twofold triple systems into twofold (K4 − e)-designs Yanxun Chang, Tao Feng 1,2 Institute of Mathematics Beijing Jiaotong University Beijing, P. R. China
Giovanni Lo Faro, Antoinette Tripodi 3,4 Department of Mathematics and Computer Science University of Messina Messina, Italy
Abstract In this note, we show that there exists a metamorphosis of a simple twofold triple system of order v into a twofold (K4 − e)-design of order v if and only if v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 10. Keywords: twofold triple system, twofold (K4 − e)-design, metamorphosis
1 2 3 4
Email: [email protected] (Y. Chang), [email protected] (T. Feng). Supported by NSFC Grant 61071221 (Y. Chang) and NSFC Grant 10901016 (T. Feng). Email: [email protected] (G. Lo Faro), [email protected] (A. Tripodi). Supported by P.R.A. and I.N.D.A.M.(G.N.S.A.G.A.) (G. Lo Faro).
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.039
218
1
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
Introduction
Let λ be a positive integer, H be a simple finite graph, and let λH denote the graph H with each of its edges replicated λ times. Let G be a simple finite graphs. A (λH, G)-design is a pair (X, B), where X is the vertex set of H and B is a collection of subgraphs (called blocks) of λH, such that each block is isomorphic to G, and each edge of λH is contained in exactly λ blocks of B. A (λH, G)-design (X, B) is called simple if B contains no repeated blocks. In what follows, the graph Kv denotes the complete graph with v vertices; K4 − e denotes a graph obtained from K4 by deleting one edge; [a, b, c − d] denotes the copy of K4 − e with vertices a, b, c, d and the edge cd removed. A (λKv , G)-design is called a λ-fold (K4 −e)-design of order v. A (λKv , Kk )design is called a balanced incomplete block design and denoted by (v, k, λ)BIBD. A (v, 3, 2)-BIBD is called a twofold triple system of order v. Let (X, B) be a simple twofold triple system of order v. For every x, y ∈ X, x = y, the pair {x, y} is contained in exactly two different triples, say, {x, y, z} and {x, y, w}. Any two blocks B1 , B2 ∈ B such that |B1 ∩ B2 | = 2 form a matched pair. Suppose now that there is a partition of B into |B|/2 matched pairs. If we replace the double edge {x, y} with its corresponding single edge {x, y} from a matched pair {x, y, z}, {x, y, w}, we are left with the (K4 − e) [x, y, z − w]. If C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and if the collection of deleted edges can be reassembled into a collection D of (K4 − e)s, then (X, C ∪ D) is a twofold (K4 − e)-design of order v, which is said to be a metamorphosis of the simple twofold triple system (X, B) and denoted by an M(2K3 > K4 − e, v) or more briefly by an M(v). For this metamorphosis M(v) to exist, it is necessary that both the simple twofold triple system and the twofold (K4 − e)-design of order v exist, that is, one must have v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 6. Example 1.1 We here give an example of an M(10). Take X = {0, 1, . . . , 9}. We construct a metamorphosis of a simple twofold triple system of order 10 (X, B) into a twofold (K4 − e)-design of order 10 (X, C ∪ D) as follows: C = {[0, 1, 4 − 5], [0, 2, 6 − 7], [0, 3, 4 − 9], [1, 2, 6 − 7], [1, 3, 5 − 9], [4, 5, 7 − 8], [4, 2, 8−9], [4, 6, 9−1], [5, 2, 9−3], [5, 6, 8−0], [7, 8, 0−1], [7, 3, 4−6], [7, 9, 5−6], [8, 3, 2 − 6], [8, 9, 0 − 1]} and D = {[0, 1, 2 − 3], [4, 5, 2 − 6], [7, 8, 3 − 9]}, where C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and the collection of deleted edges can be reassembled into a collection D of (K4 − e)s. Note that it is not necessary to list B explicitly since the blocks of B can be deduced from
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
219
C. For instance, [0, 1, 4 − 5] of C contributes two blocks {0, 1, 4} and {0, 1, 5} of B. Other results on problems in Design Theory and, in particular, on metamorphoses can be found in [2, 3, 6–13]; for more information, the interested reader may refer to [4] and the references therein. As the main result, we are to prove the following theorem. Theorem 1.2 There exists an M(v) if and only if v ≡ 0, 1, 6, 10 (mod 15) and v ≥ 10.
2
Preliminaries
Let H = {H1 , H2 , . . . , Hm } be a partition of a finite set X into subsets (called holes), where |Hi | = ni for 1 ≤ i ≤ m. Let Kn1 ,n2 ,...,nm be the complete multipartite graph on X with the i-th part on Hi , and G be a subgraph of Kn1 ,n2 ,...,nm . A λ-fold holey G-design is a triple (X, H, B) such that (X, B) is a (λKn1 ,n2 ,...,nm , G)-design. The hole type (or type) of the λ-fold holey Gdesign is the multiset {n1 , n2 , . . . , nm }. We use an “exponential” notation to describe hole types: the hole type g1u1 g2u2 · · · grur denotes ui occurrences of gi for 1 ≤ i ≤ r. When G is the complete graph Kk , a λ-fold holey Kk -design is said to be a (k, λ)-group divisible design (briefly (k, λ)-GDD). When G is K4 − e, a λ-fold holey G-design is called a λ-fold (K4 − e)-HD. Let (X, G, B) be a simple (3, 2)-GDD of group type {n1 , n2 , . . . , nm }. Suppose that |B| is even and there is a partition of B into |B|/2 matched pairs. If C is the collection of (K4 − e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and if the collection of deleted edges can be reassembled into a collection D of (K4 − e)s, then (X, G, C ∪ D) is a twofold (K4 − e)-HD with type {n1 , n2 , . . . , nm } which is said to be a metamorphosis of the simple (3, 2)-GDD (X, G, B). We usually denote the metamorphosis by M({n1 , n2 , . . . , nm }), or M(g1u1 g2u2 · · · grur ) if there are ui occurrences of gi in the hole type for 1 ≤ i ≤ r. An M(1v ) is actually an M(v). The following constructions are variations of standard constructions for group divisible designs in combinatorial design theory. Construction 2.1 (Weighting Construction) Suppose that (X, G, A) is a KGDD, and let ω : X −→ Z + ∪ {0} be a weight function. For every block A ∈ A, suppose that there is an M({ω(x) : x ∈ A}). Then there exists an M({ x∈G ω(x) : G ∈ G}).
220
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
Construction 2.2 (Filling Construction) Let a be a nonnegative integer. Suppose that there exists an M({g1 , g2 , . . . , gs }). If there is an M(1gi a1 ) for 1 ≤ i ≤ s − 1, then there exists an M(1v−gs (gs + a)1 ), where v = si=1 gi . If further there is an M(gs + a), then there exists an M(v + a). In order to use Constructions 2.1 and 2.2, we can show the following results by computer search. Lemma 2.1 There exists an M(g n ) for (g, n) ∈ {(5, 3), (5, 4), (10, 3), (10, 4), (15, 3), (15, 4), (15, 5)}. Lemma 2.2 There exist an M(1n h1 ) for (n, h) ∈ {(5, 4), (15, 6), (26, 10)}.
3
Main theorem
Lemma 3.1 There is no M(6). Proof. Suppose that there were an M(6). Then there were a simple twofold triple system of order 6 (X, B) whose blocks could be partitioned into 5 matched pairs, and only one K4 − e were reassembled from double edges, say, [a, b, c − d]. It means that a should occur in 6 different blocks of B: {a, b, ∗}, {a, b, ∗}, {a, c, ∗}, {a, c, ∗}, {a, d, ∗}, {a, d, ∗}. However, the number of blocks containing every given point of X in a twofold triple system of order 6 is 5, a contradiction. 2 Lemma 3.2 There exists an M(v) for v ≡ 0, 1, 6 (mod 15) and v ≥ 15. Proof. For v ≡ 0, 1, 6 (mod 15) and v ≥ 15, v can be written as v = 15n + h where n ≥ 1 and h = 0, 1, 6. When (n, h) ∈ {(1, 0), (1, 1), (1, 6), (2, 1)}, we can find an M(v) by computer search. When (n, h) = (2, 0), apply Construction 2.2 with an M(103 ) and an M(10), which exists by Lemma 2.1 and Example 1.1, respectively, to obtain an M(30). When (n, h) = (2, 6), apply Construction 2.2 with an M(126 101 ) from Lemma 2.2 and an M(10) to obtain an M(36). When n ≥ 3 and n = 6, 8, there is a (n, {3, 4, 5}, 1)-PBD (see [1]), that is, a {3, 4, 5}-GDD of type 1n . Give every point of the GDD weight 15 and apply Construction 2.1 to get an M(15n ), where the required M(15t ) for t = 3, 4, 5 come from Lemma 2.1. Then applying Construction 2.2 with an M(115 h1 ) and an M(15 + h), we obtain an M(v), where the required M(115 61 ) is from Lemma 2.2. When (n, h) ∈ {(6, 0), (6, 1), (8, 0), (8, 1)}, there exists a 3-GDD of type n/2 2 (see [5]). Give every point of the 3-GDD weight 15 and apply Construction 2.1 with an M(153 ) to get an M(30n/2 ). Then by Construction 2.2 with an
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
221
M(30 + h), we have the required M(v). When (n, h) ∈ {(6, 6), (8, 6)}, a 3GDD of type 3n−2 71 exists (see [5]). Give each point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(15n−2 351 ), where the required M(53 ) comes from Lemma 2.1. By Construction 2.2 with an M(16) and an M(36), we have the required M(v). 2 Lemma 3.3 There exists an M(v) for v ≡ 10 (mod 15) and v ≥ 10. Proof. For v ≡ 10 (mod 15) and v ≥ 10, v can be written as v = 5(3n + 2) where n ≥ 0. When n = 0, the conclusion follows by Example 1.1. When n = 1, we can find an M(25) by computer search. When n ≥ 2 is even, a 3-GDD of type 2(3n+2)/2 exists (see [5]). Give every point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(10(3n+2)/2 ). By Construction 2.2 with an M(10), there exists an M(v). When n ≥ 5 is odd, there is a 3-GDD of type 3n−1 51 (see [5]). Give every point of the 3-GDD weight 5 and apply Construction 2.1 to get an M(15n−1 251 ), where the required M(53 ) comes from Lemma 2.1. Apply Construction 2.2 with an M(15) and an M(25) to get an M(v). When n = 3, there exists a 4-GDD of type 34 (it is equivalent to two mutually orthogonal Latin squares of order 3), and deleting a point of the fourth group, we obtain a {3, 4}-GDD of type 33 21 . Give every point of the GDD weight 5 and apply Construction 2.1 to get an M(153 101 ), where the required M(53 ) and M(54 ) come from Lemma 2.1. Then by Construction 2.2 with an M(10) and an M(15), there exists an M(55). 2 Combing the results of Lemmas 3.1, 3.2 and 3.3, we complete the proof of Theorem 1.2.
References [1] Abel, R.J.R., F.E. Bennett, M. Grig, PBD-closures, CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds) CRC Press (2012), 247–255. [2] Amato, A., M. Gionfriddo, L. Milazzo, 2-regular equicolourings for P(4)-designs, Discrete Math. 312 (2012), 2252–2261. [3] Berardi, L., M.Gionfriddo , R. Rota, Balanced and strongly balanced P(k)designs, Discrete Math. 312 (2012), 633–636. [4] Billington, E. J., Metamorphoses of Graph Designs, URL: http://users.monash.edu.au/ accmcc/talks/TalkBillington.pdf.
222
Y. Chang et al. / Electronic Notes in Discrete Mathematics 40 (2013) 217–222
[5] Colbourn, C. J., D. G. Hoffman, and R. Rees, A new class of group divisible designs with blocks size three, J. Combin. Theory (A), 59 (1992), 73–89. [6] Chang, Y., G. Lo Faro and A. Tripodi, The spectrum of M eta(K4 − e > K3 + e, λ) with any λ, Utilitas Math. 72 (2007), 3–22. [7] Chang, Y., and G. Lo Faro, Intersection numbers of Kirkman triple systems, J. Combin. Theory (A), 2 (1999), 348–361. [8] Chang, Y., and G. Lo Faro, The flower intersection problem for Kirkman triple systems, J. Stat. Plann. and Inf., 110 (1-2)(2003), 159–177. [9] Chang, Y., G. Lo Faro and G. Nordo, The fine structures of three latin squares, Journal of Combinatorial Designs, 14 (2)(2006), 859–110. [10] Chang, Y., G. Lo Faro and A. Tripodi, Determining the spectrum of M eta(K4 + e > K4 , λ) for any λ, Discrete Math., 308 (2008), 439–456. [11] Gionfriddo, M., S. Kucukcifci, L. Milazzo , Balanced and Strongly Balanced 4-Kite designs, Utilitas Mathematica 91 (2013). [12] Lindner, C.C., and A. Tripodi, The metamorphosis of K4 − e designs into maximum packings of Kn with 4-cycles, Ars Combinatoria, 75 (2005), 333–349. [13] Lo Faro, G., A. Tripodi, The Doyen-Wilson theorem for kite systems, Discrete Math., 306 (2006), 2695–2701.