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The Doyen—Wilson theorem extended to 5-cycles
JOURNAL OF COMBINATORIAL THEORY, Series A 68, 2 1 8 - 2 2 5 (1994)
Note The Doyen-Wilson Theorem Extended to 5-Cycles DARRYN
E. BRYANT
Centre for Combinatorio, Department of Mathematics, The University of Queensland, Qld 4072, Australia AND
C. A. RODGER Department of Discrete and Statistical Sciences, Auburn University,Auburn, Alabama 36849-5307 Communicated by the Managing Editors R e c e i v e d S e p t e m b e r 1, 1992
In this paper it is proved that any 5-cycle system of order u can be embedded in a 5-cycle system of order v iff v > 3u/2 and v --- 1 or 5 (rood 10). © 1994Academic Press, Inc.
1. INTRODUCTION Let ~m = {0, 1 . . . . , m - 1}. A n m-cycle is a g r a p h (v0, V l , . . . , u~_ 1) with vertex set {viii ~ E m} and edge set {{vi, Vi+a}]i ~ 7/m} (reducing the subscript m o d u l o m). A n m-cycle system of a graph G is an o r d e r e d pair (V, C), w h e r e V is the vertex set of G and C is a set of m-cycles, the edges of which partition the edges of G. A n m-cycle system (of order n) is an m-cycle system of K n. A partial m-cycle system of order n is an m-cycle system of a subgraph of Kn. In this paper, we are interested in the e m b e d d i n g p r o b l e m for m-cycle systems. A (partial) m-cycle system (V, C) is said to be embedded in the m-cycle system (W, P ) if V c W and C c P. In 1973, D o y e n and Wilson o b t a i n e d the following result for e m b e d d i n g 3-cycle systems, also k n o w n as Steiner triple systems. THEOREM 1.1 [3]. A Steiner triple system of order u can be embedded in a Steiner triple system of order v > u iff v >__2u + 1 and v =- 1 or 3 (rood 6). 218 0 0 9 7 - 3 1 6 5 / 9 4 $6.00 Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.
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This complete solution is in contrast to what is known about the embedding of partial Steiner triple systems. After earlier results by Treash [15] and Lindner [6], the best result to date, by Andersen Hilton, and Mendelsohn [1], shows that any partial 3-cycle system of order u can be embedded in a 3-cycle system of order v for any v > 4u + 1 and u - 1 or 3 (mod 6). The best possible result would be to obtain this result with the lower bound on u replaced by u > 2u + 1 (this bound has been achieved for partial triple systems with index divisible by 4 [4]). More recently, the embedding of m-cycle systems and of partial m-cycle systems has been considered. The following lemma provides a lower bound on the order of the containing m-cycle system when m is odd. LEMMA 1.2 [12]. I f m is odd and if an m-cycle system of order u has been embedded in an m-cycle system of order v > u, then v > (m + 1 ) u / ( m - 1). After some previous results [10, 11], the best embeddings for partial m-cycle systems have now reduced the size of the containing system to u = 2 n m + 1 and (2n + 1)m [8] if m is even or odd, respectively (see [7] for a survey and [9] for a summary of the best embeddings). It turns out that when looking for extensions of the Doyen-Wilson theorem to cycles of length greater than 3, one can obtain some such theorems [2] in the case where m is even by using a result by Sotteau [13]. However, such extensions are much more difficult to find for m-cycle systems when m is odd. In this paper we find such an extension in the case where m = 5, showing that the correct lower bound on the size of the containing square is c, > 3 u / 2 , as suggested by Lemma 1.2 (see Theorem 3.2).
2. NOTATION AND PRELIMINARY RESULTS Let G c denote the complement of G. If G and H are two graphs then let G U H be the graph with vertex set V ( G U H ) = V ( G ) U V ( H ) and edge set E ( G u H ) = E ( G ) U E ( H ) . If V ( G ) N V ( H ) = Q, then let G v H be the graph with V(G v H ) = V ( G ) u V ( H ) and E ( G V H ) = E ( G ) U E ( H ) U {{g, h}lg ~ V(G), h ~ V(H)}. Define li - J l x = min{li - j l , x - li -jl}. If Do, D 1 _ {1,2 . . . . ,[x/2J} and S _c Yx, then define the graph {D0, S, D1}~ to be the graph with vertex set 2~ × 2 2 and edge set {{(i, 0),(j, 0)}l l i - j l x ~ Do} O {{(i, 0), (j, 1)}1i - j(mod x) ~ S} U {{(i, 1),(j, 1)}1 li - j l ~ ~ D1}. The following lemma is a particular case of a result of Stern and Lenz. LEMMA 2.1 [14]. Let D c_ {1, 2 , . . . , [x/21} and S c_ Z , . I f either tSI > 1 or x / 2 ~ D then { D, S, D )x has a 1-factorization.
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We need several building blocks to obtain the required embedding. LEMMA 2.2.
For any x > 2, there exists a 5-cycle system of K~ v C2x.
Proof. The proof is by induction on x. It is easy to see that there exists a 5-cycle system of K~ v C 4. So assume that there exists a 5-cycle system of K~ v C2,, the four vertices in K~ being 001, % , % , and %, and the cycle Cz, being (v o, v 1. . . . . v2,_1). Let the three 5-cycles containing U2x_1 be (vo, v 2 x _ l , % , a l , az), (/02x_2, V2x_l, oo2,a3, a4) , and (003,v2x-1,004, as, a6), where { a l , . . . , a 6} _ V(K~ U Czx). We add two new vertices V2x and V2x+l and replace these three 5-cycles with (v o, v2~+l, %, a 1, a2), (v2~_z, v2x_l, 002, a3, a4), (003' U2x' 004' as, a6), (001' U2x' V2x+ 1' 004' U2x-1 )' a n d (002' V2x' V2x_1,%,v2x+l) to obtain a 5-cycle system of K~ V C2x+2 , the cycle C2x+2 being (v o, V l , . . . , V2x+l). I A 2-factor of a graph is said to be even if each component is a cycle of even length. Then we immediately have the following result. COROLLARY 2.3. system Of K~ V T. LEMMA 2.4. Kc).
I f T is an even 2-factor then there exists a 5-cycle
For any x > 3, there exists a 5-cycle system of K~ V (C x U
Proof. Let the vertices of K~ be 001 and 002, let the cycle C, be (Vo, V l , . . . , G _ l ) , and let the vertices of K~ be Wo, W l , . . . , w ~ _ p Then {(00Dvi, vi+1,002, wi)li E Zx}, reducing subscripts modulo x, is a set of 5-cycles forming the required 5-cycle system. | LEMMA 2.5. I f y < (2X -- 2 ) / 3 then there exists a 3y-regular graph G with vertex set 7/, × 7/2 such that there exists a 5-cycle system o f Ky v G and such that any one of the following is satisfied: (1) G c has a 1-factorization; (2) /f y < (2x - 4 ) / 3 and 5Ix then G ~ - ({x/5, O,{x/5})~ has a 1-factorization ; (3) if y < (2x - 6 ) / 3 and 5Ix then G c - ( { x / 5 , 2 x / 5 } , 0 , {x / 5, 2 x / 5} )x has a 1-factorization. Proof Let the vertex set of G be Z~ × Z 2 and the vertex set of Ky be {%,...,%}. Let h = [ ( x - 1 ) / 2 ] . Let S 1 = O, S 2 = {x/5}, S 3 = { x / 5 , 2x/5}, D 1 = {1, 2 . . . . . h}, O 2 = D 1 \ $2, and D 3 = D 1 \ S 3. To obtain property (i), 1 < i < 3, select min{y, h - i + 1} elements of D i and select min{y, h - i + 1} elements of S = Z,, and pair the selected elements of Di to the selected elements of S arbitrarily, and then assign a
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vertex ~ to the jth such pair, 1 _
I LEMMA 2.6.
There exists a 3-regular graph G on 10 vertices such that
(a) there exists a 5-cycle system of K~ v G, and (b) G c has a 1-factorization. Proof Let V ( G ) = Z 5 × 7/2 and E ( G ) = (Q, {0,1, 4}, Q)5. Then by Lemma 2.1, G c has a 1-factorization, so (b) is satisfied. Let the vertices of K~ be u, v, and w. Then the set of 5-cycles {(u, (0, 0), (0, 1), (1, 0), (2, 1)), (v, (1, 1), (2, 0), (2, 1), (3, 0)), ( w , (3, 1), (3, 0), (4, 1), (4, 0)), ( u , (2, 0), (3, 1), v, (4, 1)), (u, (3, 1), (4, 0), v, (0, 1)), (u, (1, 1), (0, 0), w, (3, 0)), (u, (4, 0), (0, 1), w, (1, 0)), (v, (0, 0), (4, 1), w, (2, 0)), (v', (1, 0), (1, 1), w, (2, 1))} partition the edges of K~ V G, as required by (a). |
Finally, we need the following result for one family of small embeddings. A graph G is overfull if e(G) > A(G)[v/2], where e(G) = IE(G)t, u(G) = IV(G)I and A(G) is the maximum degree of G. THEOREM 2.7 [5]. The complete multipartite graph has chromatic index equal to the maximum degree iff it is not overfull.
3. THE MAIN RESULT
To prove the main result, we begin by finding the smallest embeddings in Proposition 3.1, then use these to inductively prove Theorem 3.2. PROPOSITION 3.1. Let (V, P ) be a 5-cycle system of order u. Let 3u > v > 3 u / 2 with v =- 1 or 5 (rood 10). Then (V, P ) can be embedded in a 5-cycle system of order v. 582a/68/1-15
222
NOTE
Proof. W e c o n s i d e r several cases in turn. Case 1. u - 1 ( m o d l 0 ) and v - 5 ( m o d l 0 ) . Let u = 2 0 s + 10e+ 1 with e ~ { 0 , 1 } a n d u > 11 (so s > 0 a n d if e = 0 t h e n s> 1).Thenwe can write v = 3 0 s + 2 0 e + 5 + 1 0 t , w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 9. Note that this lower b o u n d o n v implies that t > 0 a n d if v > 4 u - 9 t h e n v > 3 u . W e add the set W = Zs(s+t+~+2 × Z 2 of n e w vertices to V. P a r t i t i o n the vertices of V into o n e set of size 4t + 2e + 1 a n d 5s - t + 2e sets of size 4. Since v < 4 u - 9, we c a n apply L e m m a 2.5(1) with x = 5(s + t +e)+2 and y =4t+2e+ 1 to the set of 4 t + 2 e + 1 vertices. By L e m m a 2.5(1), the r e m a i n i n g edges j o i n i n g pairs of vertices in W form a ( 2 x - 1 - 3 y ) - r e g u l a r graph that has a 1-factorization. Since 2 x - 1 3y = 10s - 2 t + 4e is even, we can pair off the i-factors to form 5s t + 2e even 2-factors. T h e 5s - t + 2e even 2-factors can now be arbitrarily m a t c h e d with the 5s - t + 2e sets of size 4, a n d t h e n Corollary 2.3 can b e a p p l i e d 5s - t + 2e times to c o m p l e t e the e m b e d d i n g . Case 2. u -- 5 (rood 10) and v - 1 (mod 10). This follows Case 1 very closely. Let u = 20s - 10e + 5 with e ~ {0, 1} a n d u > 5 (so s > 0 a n d if e = 1 t h e n s > 1). T h e n we can write v = 30s - 10e + 11 + 10t, w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 9. A g a i n v > 3u/2 implies that t>0andif v>4u-9thenv>3u. A d d the set W = Z(5,+t~+3 x 7/2 of n e w vertices to V. P a r t i t i o n the vertices of V into o n e set of size 4t + 2e + 1 a n d 5s - t - 3e + 1 sets of size 4. Since v < 4 u - 9, we can apply L e m m a 2.5(1) with x = 5(s + t) + 3 and y=4t +2e+ 1 to the set of 4 t + 2 e + 1 vertices. By L e m m a 2.5(1), the r e m a i n i n g edges j o i n i n g pairs of vertices in W form a (2x - 1 - 3 y ) - r e g u l a r graph that has a l-factorization. Since 2 x - 1 3y = 10s - 2t - 6e - 2 is even, the 1-factors can b e p a i r e d off a n d the r e s u l t i n g even 2-factors t o g e t h e r with the 5s - t - 3e + 1 sets of size 4 can b e u s e d in Corollary 2.3 to c o m p l e t e the e m b e d d i n g . Case 3. u - - 1 ( m o d 1 0 ) and v - - 1 ( m o d 1 0 ) . Let u = 2 0 s 10e + 1 with e ~ { 0 , 1 } a n d u > 11 (so s > 1). T h e n we can write v = 3 0 s 20e + 11 + 10t, w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 13. A g a i n v > 3u/2 implies that t > 0 a n d if v > 4 u - 13 t h e n v > 3u. A d d the set W = 7/5(s+t-~+~) × Z2 of n e w vertices to V. P a r t i t i o n the vertices into o n e set of size 2, o n e set of size 4t - 2e + 3, a n d 5s - t 2e - 1 sets of size 4. Since v < 4 u - 13, we can apply L e m m a 2.5(2) with x=5(s+t-e + 1) a n d y = 4 t - 2 e + 3 to the set of 4 t - 2 e +3 vertices. By L e m m a 2.5(2), the edges ( { x / 5 } , Q, {x/5})x are still r e m a i n ing: the vertices of V in the set of size 2 t o g e t h e r with the edges in ( { x / 5 } , •, O)x c a n be used in L e m m a 2.4 a n d the edges in ( Q , Q, {x/5})x
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a r e a set of v e r t e x disjoint 5-cycles. Finally, the r e m a i n i n g e d g e s j o i n i n g vertices in W f o r m a ( 2 x - 3 - 3 y ) - r e g u l a r g r a p h which by L e m m a 2.5(2) has a 1-factorization. Since 2 x - 3 - 3 y = 10s-2t-4e-2, these e d g e s t o g e t h e r with t h e 5s - t - 2 e - 1 sets of size 4 can b e u s e d in C o r o l l a r y 2.3 as in t h e p r e v i o u s cases to c o m p l e t e t h e e m b e d d i n g .
Case 4. u - 5 (rood 10) and v = 5 (rood 10). In this case, t h e smallest e m b e d d i n g w h e n u = 15 ( m o d 20) proves to b e different from any o t h e r case, so w e b e g i n with it. L e t u = 20s + 15 a n d u = 30s + 25, w h e r e s > 0. A d d the set W = Z10 × 7/~+ 1 of n e w vertices to V. P a r t i t i o n t h e vertices o f V into o n e set o f size 3 a n d 5s + 3 sets of size 4. F o r e a c h i ~ Z~+ 1 let G i b e a 3 - r e g u l a r g r a p h on t h e v e r t e x set 210 × 7/i, as d e s c r i b e d in L e m m a 2.6. A p p l y L e m m a 2.6 to t h e set o f 3 vertices of V a n d Gi for e a c h i ~ Zs+ 1. F r o m L e m m a 2.6(b), G/~ has a 1-factorization, a n d t o g e t h e r t h e 1-factorizations o f G o, G 1 , . . . , G s f o r m six 1-factors on the v e r t e x set 7/10 x 2~+1. W e use t h e s e six 1-factors t o g e t h e r with t h e r e m a i n i n g e d g e s which, since t h e y i n d u c e a c o m p l e t e (s + 1)-partite g r a p h with 10 vertices in e a c h part, by T h e o r e m 2.7 also have a 1-factorization. So a l t o g e t h e r t h e s e 10s + 6 1-factors can b e u s e d to f o r m even 2-factors a n d c o m b i n e d with the 5s + 3 sets o f 4 vertices of V using L e m m a 2.2 to c o m p l e t e the e m b e d d i n g . So now w e can let u = 20s - 10e + 5 with e ~ {0, 1} a n d u > 5 (so s>_0andife= l a n d s > 1) a n d u = 3 0 s 1 0 e + 1 5 + 10t, w h e r e t is any i n t e g e r such t h a t 3 u / 2 < u _< 4u - 5. I n this case, v >>_3 u / 2 implies t h a t t > - 1 , a n d if t = - 1 t h e n e = 1. T h e case w h e r e t = - 1 a n d e -- 1 is t h e s m a l l e s t e m b e d d i n g just c o n s i d e r e d . So a g a i n w e n e e d only c o n s i d e r t > 0. A l s o , if v > 4u - 5 t h e n v > 3u. A d d t h e set 7/S(s +t+ l) X 7/2 of n e w vertices to V. P a r t i t i o n t h e vertices of V into o n e set of size 4 t + 1 + 2 e a n d 5s - t - 3e + 1 sets of size 4. Since u < 4 u - 5 , w e can a p p l y L e m m a 2.5(3) with x = 5 ( s + t + 1) a n d y = 4 t + 1 + 2e to t h e set o f 4 t + 1 + 2 e vertices. T h e e d g e s in ({s + t + 1},Q,{s + t + 1})x a n d ({2(s + t + 1)},Q,{2(s + t + 1)})x a r e e a c h sets of v e r t e x disjoint 5-cycles. A n d t h e only e d g e s j o i n i n g vertices in W t h a t r e m a i n can, by L e m m a 2.5(3), b e p a r t i t i o n e d into 2 x - 1 - 3y 4 = 10s - 2 t - 6e + 2 1-factors. A s b e f o r e , t h e s e 1-factors can b e u s e d to f o r m 5s - t - 3e + 1 even 2-factors which t o g e t h e r with the 5s - t + 3e + 1 sets of 4 v e r t i c e s o f V a n d using L e m m a 2.2 c o m p l e t e t h e embedding. | W e a r e n o w r e a d y for t h e e x t e n s i o n o f t h e D o y e n - W i l s o n t h e o r e m . THEOREM 3.2. Let (V, P ) be a 5-cycle system of order u. Then (V, P ) can be embedded in a 5-cycle system o f order u > u iff v > 3 u / 2 .
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NOTE
Proof. T h e r e q u i r e m e n t that v > 3 u / 2 follows directly from L e m m a 1.2. T h e p r o o f of the sufficiency is by i n d u c t i o n o n v. Clearly we c a n a s s u m e t h a t u > 5. By P r o p o s i t i o n (3.1), the e m b e d d i n g is possible if 3 u / 2 < v < 3u. Now let v - 1 or 5 ( m o d 1 0 ) a n d s u p p o s e that for all r - - 1 or 5 (mod 10) with 3 u / 2 < r < v a n d v > 3u, (V, P ) c a n be e m b e d d e d in a 5-cycle system of o r d e r r. Since v >__ 3u + 1, the greatest i n t e g e r less t h a n 2 v / 3 that is c o n g r u e n t to 1 or 5 (mod 10) is certainly at least 2 u - 6. Also, the least i n t e g e r g r e a t e r t h a n 3 u / 2 that is c o n g r u e n t to 1 or 5 ( m o d l 0 ) is at most (3u + 9 ) / 2 (recall that u - 1 or 5 (rood 10)). T h e r e f o r e , if u > 21 t h e n t h e r e exists a n i n t e g e r s -= 1 or 5 (rood 10) with 3 u / 2 < s < 2 v / 3 . Similarly, if u = 15 t h e n v > 51, if u = 11 t h e n v > 35, a n d if u = 5 t h e n v >__ 21, so in every case such a n i n t e g e r s can b e found. I n any case, let t b e the greatest i n t e g e r with 3 u / 2 < t < 2 v / 3 a n d t --- 1 or 5 (rood 10). T h e n by i n d u c t i o n (V, P ) can be e m b e d d e d in a 5-cycle system of o r d e r t, which by P r o p o s i t i o n 3.1 c a n b e e m b e d d e d in a 5-cycle system of o r d e r v. II
ACKNOWLEDGMENTS The second author (C.A.R.) thanks The University of Queensland and the Ethel Raybould Fellowship for support while this research was being done. This research was also supported by the Australian Research Council (DEB) and by the NSF under Grant DMS-9024645 (C.A.R.).
REFERENCES 1. L. D. ANDERSEN,A. J. W. HILTON, AND E. MENDELSOHN, Embedding partial Steiner triple systems, Proc. London Math. Soc. 41 (1980), 557-576. 2. D. E. BRYANTAND C. A. RODGER,Embedding even cycle systems, in preparation. 3. J. DOYEN AND R. M. WILSON, Embeddings of Steiner triple systems, Discrete Math 5 (1973), 229-239. 4. A, J. W. HILTONAND C. A. RODGER, The embedding of partial triple systems when 4 divides J. Combin. Theory Ser. A 56 (1991), 109-137. 5. D. G. HOFFMANAND C. A. RODGER, The chromatic index of complete multipartite graphs, J. Graph Theory 16 (1992), 159-163. 6. C. C. LIN~NER, A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6n + 3, J. Combin. Theory Ser..4 18 (1975), 349-351. 7. C. C. LINDNER AND C. A. RODGER, Decomposition into cycles. II. Cycle systems, in "Contemporary Design Theory: A Collection of Surveys" (J. H. Dinitz and D. R. Stinson, Eds.) Wiley, New York, 1992.
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8. C. C. LINDNER AND C. A. RODGER, A partial m = (2k + 1)-cycle system of order n can be embedded in an m-cycle system of order (2n + 1)m, Discrete Math., 117 (1993), 151-159. 9. C. C. LrZ~DNER AND C. A. RODCER, Embedding directed and undirected partial cycle systems of index A > 1, J. Combin. Designs 2 (1993), 113-123. 10. C. C. LrNDNER, C. A. RODOER, AND D. R. STINSON, Embedding cycle systems of even length, J. Combin. Math. Combin, Comput. 3 (1988), 65-69. 11. C. C. LINDNER C. A. RODCER, AYD D. R. STINSON, Small embeddings for partial cycle systems of odd length, Discrete Math. 80 (1990), 273-280. 12. C. A. RODGER, Problems on cycle systems of odd length, Congr. Numer. 61 (1988), 5-22. . 13. D. SoTrEAU, Decompositions of Km, n(Km. n) into cycles (circuits) of length 2k, J. Comb~n. Theory Ser. B 30 (1981), 75-81. 14. G. STERN AND H. LENZ, Steiner triple systems with given subspaces; another proof of the Doyen-Wilson theorem, Boll. Un. Mat. Ital. A (5) 17 (1980), 109-114. 15. C. TREASH, The completion of finite incomplete Steiner triple systems with application to loop theory, J. Combin. Theory Ser. A. 10 (1971), 250-265.
Note The Doyen-Wilson Theorem Extended to 5-Cycles DARRYN
E. BRYANT
Centre for Combinatorio, Department of Mathematics, The University of Queensland, Qld 4072, Australia AND
C. A. RODGER Department of Discrete and Statistical Sciences, Auburn University,Auburn, Alabama 36849-5307 Communicated by the Managing Editors R e c e i v e d S e p t e m b e r 1, 1992
In this paper it is proved that any 5-cycle system of order u can be embedded in a 5-cycle system of order v iff v > 3u/2 and v --- 1 or 5 (rood 10). © 1994Academic Press, Inc.
1. INTRODUCTION Let ~m = {0, 1 . . . . , m - 1}. A n m-cycle is a g r a p h (v0, V l , . . . , u~_ 1) with vertex set {viii ~ E m} and edge set {{vi, Vi+a}]i ~ 7/m} (reducing the subscript m o d u l o m). A n m-cycle system of a graph G is an o r d e r e d pair (V, C), w h e r e V is the vertex set of G and C is a set of m-cycles, the edges of which partition the edges of G. A n m-cycle system (of order n) is an m-cycle system of K n. A partial m-cycle system of order n is an m-cycle system of a subgraph of Kn. In this paper, we are interested in the e m b e d d i n g p r o b l e m for m-cycle systems. A (partial) m-cycle system (V, C) is said to be embedded in the m-cycle system (W, P ) if V c W and C c P. In 1973, D o y e n and Wilson o b t a i n e d the following result for e m b e d d i n g 3-cycle systems, also k n o w n as Steiner triple systems. THEOREM 1.1 [3]. A Steiner triple system of order u can be embedded in a Steiner triple system of order v > u iff v >__2u + 1 and v =- 1 or 3 (rood 6). 218 0 0 9 7 - 3 1 6 5 / 9 4 $6.00 Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.
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219
This complete solution is in contrast to what is known about the embedding of partial Steiner triple systems. After earlier results by Treash [15] and Lindner [6], the best result to date, by Andersen Hilton, and Mendelsohn [1], shows that any partial 3-cycle system of order u can be embedded in a 3-cycle system of order v for any v > 4u + 1 and u - 1 or 3 (mod 6). The best possible result would be to obtain this result with the lower bound on u replaced by u > 2u + 1 (this bound has been achieved for partial triple systems with index divisible by 4 [4]). More recently, the embedding of m-cycle systems and of partial m-cycle systems has been considered. The following lemma provides a lower bound on the order of the containing m-cycle system when m is odd. LEMMA 1.2 [12]. I f m is odd and if an m-cycle system of order u has been embedded in an m-cycle system of order v > u, then v > (m + 1 ) u / ( m - 1). After some previous results [10, 11], the best embeddings for partial m-cycle systems have now reduced the size of the containing system to u = 2 n m + 1 and (2n + 1)m [8] if m is even or odd, respectively (see [7] for a survey and [9] for a summary of the best embeddings). It turns out that when looking for extensions of the Doyen-Wilson theorem to cycles of length greater than 3, one can obtain some such theorems [2] in the case where m is even by using a result by Sotteau [13]. However, such extensions are much more difficult to find for m-cycle systems when m is odd. In this paper we find such an extension in the case where m = 5, showing that the correct lower bound on the size of the containing square is c, > 3 u / 2 , as suggested by Lemma 1.2 (see Theorem 3.2).
2. NOTATION AND PRELIMINARY RESULTS Let G c denote the complement of G. If G and H are two graphs then let G U H be the graph with vertex set V ( G U H ) = V ( G ) U V ( H ) and edge set E ( G u H ) = E ( G ) U E ( H ) . If V ( G ) N V ( H ) = Q, then let G v H be the graph with V(G v H ) = V ( G ) u V ( H ) and E ( G V H ) = E ( G ) U E ( H ) U {{g, h}lg ~ V(G), h ~ V(H)}. Define li - J l x = min{li - j l , x - li -jl}. If Do, D 1 _ {1,2 . . . . ,[x/2J} and S _c Yx, then define the graph {D0, S, D1}~ to be the graph with vertex set 2~ × 2 2 and edge set {{(i, 0),(j, 0)}l l i - j l x ~ Do} O {{(i, 0), (j, 1)}1i - j(mod x) ~ S} U {{(i, 1),(j, 1)}1 li - j l ~ ~ D1}. The following lemma is a particular case of a result of Stern and Lenz. LEMMA 2.1 [14]. Let D c_ {1, 2 , . . . , [x/21} and S c_ Z , . I f either tSI > 1 or x / 2 ~ D then { D, S, D )x has a 1-factorization.
220
NOTE
We need several building blocks to obtain the required embedding. LEMMA 2.2.
For any x > 2, there exists a 5-cycle system of K~ v C2x.
Proof. The proof is by induction on x. It is easy to see that there exists a 5-cycle system of K~ v C 4. So assume that there exists a 5-cycle system of K~ v C2,, the four vertices in K~ being 001, % , % , and %, and the cycle Cz, being (v o, v 1. . . . . v2,_1). Let the three 5-cycles containing U2x_1 be (vo, v 2 x _ l , % , a l , az), (/02x_2, V2x_l, oo2,a3, a4) , and (003,v2x-1,004, as, a6), where { a l , . . . , a 6} _ V(K~ U Czx). We add two new vertices V2x and V2x+l and replace these three 5-cycles with (v o, v2~+l, %, a 1, a2), (v2~_z, v2x_l, 002, a3, a4), (003' U2x' 004' as, a6), (001' U2x' V2x+ 1' 004' U2x-1 )' a n d (002' V2x' V2x_1,%,v2x+l) to obtain a 5-cycle system of K~ V C2x+2 , the cycle C2x+2 being (v o, V l , . . . , V2x+l). I A 2-factor of a graph is said to be even if each component is a cycle of even length. Then we immediately have the following result. COROLLARY 2.3. system Of K~ V T. LEMMA 2.4. Kc).
I f T is an even 2-factor then there exists a 5-cycle
For any x > 3, there exists a 5-cycle system of K~ V (C x U
Proof. Let the vertices of K~ be 001 and 002, let the cycle C, be (Vo, V l , . . . , G _ l ) , and let the vertices of K~ be Wo, W l , . . . , w ~ _ p Then {(00Dvi, vi+1,002, wi)li E Zx}, reducing subscripts modulo x, is a set of 5-cycles forming the required 5-cycle system. | LEMMA 2.5. I f y < (2X -- 2 ) / 3 then there exists a 3y-regular graph G with vertex set 7/, × 7/2 such that there exists a 5-cycle system o f Ky v G and such that any one of the following is satisfied: (1) G c has a 1-factorization; (2) /f y < (2x - 4 ) / 3 and 5Ix then G ~ - ({x/5, O,{x/5})~ has a 1-factorization ; (3) if y < (2x - 6 ) / 3 and 5Ix then G c - ( { x / 5 , 2 x / 5 } , 0 , {x / 5, 2 x / 5} )x has a 1-factorization. Proof Let the vertex set of G be Z~ × Z 2 and the vertex set of Ky be {%,...,%}. Let h = [ ( x - 1 ) / 2 ] . Let S 1 = O, S 2 = {x/5}, S 3 = { x / 5 , 2x/5}, D 1 = {1, 2 . . . . . h}, O 2 = D 1 \ $2, and D 3 = D 1 \ S 3. To obtain property (i), 1 < i < 3, select min{y, h - i + 1} elements of D i and select min{y, h - i + 1} elements of S = Z,, and pair the selected elements of Di to the selected elements of S arbitrarily, and then assign a
NOTE
221
vertex ~ to the jth such pair, 1 _
I LEMMA 2.6.
There exists a 3-regular graph G on 10 vertices such that
(a) there exists a 5-cycle system of K~ v G, and (b) G c has a 1-factorization. Proof Let V ( G ) = Z 5 × 7/2 and E ( G ) = (Q, {0,1, 4}, Q)5. Then by Lemma 2.1, G c has a 1-factorization, so (b) is satisfied. Let the vertices of K~ be u, v, and w. Then the set of 5-cycles {(u, (0, 0), (0, 1), (1, 0), (2, 1)), (v, (1, 1), (2, 0), (2, 1), (3, 0)), ( w , (3, 1), (3, 0), (4, 1), (4, 0)), ( u , (2, 0), (3, 1), v, (4, 1)), (u, (3, 1), (4, 0), v, (0, 1)), (u, (1, 1), (0, 0), w, (3, 0)), (u, (4, 0), (0, 1), w, (1, 0)), (v, (0, 0), (4, 1), w, (2, 0)), (v', (1, 0), (1, 1), w, (2, 1))} partition the edges of K~ V G, as required by (a). |
Finally, we need the following result for one family of small embeddings. A graph G is overfull if e(G) > A(G)[v/2], where e(G) = IE(G)t, u(G) = IV(G)I and A(G) is the maximum degree of G. THEOREM 2.7 [5]. The complete multipartite graph has chromatic index equal to the maximum degree iff it is not overfull.
3. THE MAIN RESULT
To prove the main result, we begin by finding the smallest embeddings in Proposition 3.1, then use these to inductively prove Theorem 3.2. PROPOSITION 3.1. Let (V, P ) be a 5-cycle system of order u. Let 3u > v > 3 u / 2 with v =- 1 or 5 (rood 10). Then (V, P ) can be embedded in a 5-cycle system of order v. 582a/68/1-15
222
NOTE
Proof. W e c o n s i d e r several cases in turn. Case 1. u - 1 ( m o d l 0 ) and v - 5 ( m o d l 0 ) . Let u = 2 0 s + 10e+ 1 with e ~ { 0 , 1 } a n d u > 11 (so s > 0 a n d if e = 0 t h e n s> 1).Thenwe can write v = 3 0 s + 2 0 e + 5 + 1 0 t , w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 9. Note that this lower b o u n d o n v implies that t > 0 a n d if v > 4 u - 9 t h e n v > 3 u . W e add the set W = Zs(s+t+~+2 × Z 2 of n e w vertices to V. P a r t i t i o n the vertices of V into o n e set of size 4t + 2e + 1 a n d 5s - t + 2e sets of size 4. Since v < 4 u - 9, we c a n apply L e m m a 2.5(1) with x = 5(s + t +e)+2 and y =4t+2e+ 1 to the set of 4 t + 2 e + 1 vertices. By L e m m a 2.5(1), the r e m a i n i n g edges j o i n i n g pairs of vertices in W form a ( 2 x - 1 - 3 y ) - r e g u l a r graph that has a 1-factorization. Since 2 x - 1 3y = 10s - 2 t + 4e is even, we can pair off the i-factors to form 5s t + 2e even 2-factors. T h e 5s - t + 2e even 2-factors can now be arbitrarily m a t c h e d with the 5s - t + 2e sets of size 4, a n d t h e n Corollary 2.3 can b e a p p l i e d 5s - t + 2e times to c o m p l e t e the e m b e d d i n g . Case 2. u -- 5 (rood 10) and v - 1 (mod 10). This follows Case 1 very closely. Let u = 20s - 10e + 5 with e ~ {0, 1} a n d u > 5 (so s > 0 a n d if e = 1 t h e n s > 1). T h e n we can write v = 30s - 10e + 11 + 10t, w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 9. A g a i n v > 3u/2 implies that t>0andif v>4u-9thenv>3u. A d d the set W = Z(5,+t~+3 x 7/2 of n e w vertices to V. P a r t i t i o n the vertices of V into o n e set of size 4t + 2e + 1 a n d 5s - t - 3e + 1 sets of size 4. Since v < 4 u - 9, we can apply L e m m a 2.5(1) with x = 5(s + t) + 3 and y=4t +2e+ 1 to the set of 4 t + 2 e + 1 vertices. By L e m m a 2.5(1), the r e m a i n i n g edges j o i n i n g pairs of vertices in W form a (2x - 1 - 3 y ) - r e g u l a r graph that has a l-factorization. Since 2 x - 1 3y = 10s - 2t - 6e - 2 is even, the 1-factors can b e p a i r e d off a n d the r e s u l t i n g even 2-factors t o g e t h e r with the 5s - t - 3e + 1 sets of size 4 can b e u s e d in Corollary 2.3 to c o m p l e t e the e m b e d d i n g . Case 3. u - - 1 ( m o d 1 0 ) and v - - 1 ( m o d 1 0 ) . Let u = 2 0 s 10e + 1 with e ~ { 0 , 1 } a n d u > 11 (so s > 1). T h e n we can write v = 3 0 s 20e + 11 + 10t, w h e r e t is any i n t e g e r such that 3u/2 < v < 4 u - 13. A g a i n v > 3u/2 implies that t > 0 a n d if v > 4 u - 13 t h e n v > 3u. A d d the set W = 7/5(s+t-~+~) × Z2 of n e w vertices to V. P a r t i t i o n the vertices into o n e set of size 2, o n e set of size 4t - 2e + 3, a n d 5s - t 2e - 1 sets of size 4. Since v < 4 u - 13, we can apply L e m m a 2.5(2) with x=5(s+t-e + 1) a n d y = 4 t - 2 e + 3 to the set of 4 t - 2 e +3 vertices. By L e m m a 2.5(2), the edges ( { x / 5 } , Q, {x/5})x are still r e m a i n ing: the vertices of V in the set of size 2 t o g e t h e r with the edges in ( { x / 5 } , •, O)x c a n be used in L e m m a 2.4 a n d the edges in ( Q , Q, {x/5})x
NOTE
223
a r e a set of v e r t e x disjoint 5-cycles. Finally, the r e m a i n i n g e d g e s j o i n i n g vertices in W f o r m a ( 2 x - 3 - 3 y ) - r e g u l a r g r a p h which by L e m m a 2.5(2) has a 1-factorization. Since 2 x - 3 - 3 y = 10s-2t-4e-2, these e d g e s t o g e t h e r with t h e 5s - t - 2 e - 1 sets of size 4 can b e u s e d in C o r o l l a r y 2.3 as in t h e p r e v i o u s cases to c o m p l e t e t h e e m b e d d i n g .
Case 4. u - 5 (rood 10) and v = 5 (rood 10). In this case, t h e smallest e m b e d d i n g w h e n u = 15 ( m o d 20) proves to b e different from any o t h e r case, so w e b e g i n with it. L e t u = 20s + 15 a n d u = 30s + 25, w h e r e s > 0. A d d the set W = Z10 × 7/~+ 1 of n e w vertices to V. P a r t i t i o n t h e vertices o f V into o n e set o f size 3 a n d 5s + 3 sets of size 4. F o r e a c h i ~ Z~+ 1 let G i b e a 3 - r e g u l a r g r a p h on t h e v e r t e x set 210 × 7/i, as d e s c r i b e d in L e m m a 2.6. A p p l y L e m m a 2.6 to t h e set o f 3 vertices of V a n d Gi for e a c h i ~ Zs+ 1. F r o m L e m m a 2.6(b), G/~ has a 1-factorization, a n d t o g e t h e r t h e 1-factorizations o f G o, G 1 , . . . , G s f o r m six 1-factors on the v e r t e x set 7/10 x 2~+1. W e use t h e s e six 1-factors t o g e t h e r with t h e r e m a i n i n g e d g e s which, since t h e y i n d u c e a c o m p l e t e (s + 1)-partite g r a p h with 10 vertices in e a c h part, by T h e o r e m 2.7 also have a 1-factorization. So a l t o g e t h e r t h e s e 10s + 6 1-factors can b e u s e d to f o r m even 2-factors a n d c o m b i n e d with the 5s + 3 sets o f 4 vertices of V using L e m m a 2.2 to c o m p l e t e the e m b e d d i n g . So now w e can let u = 20s - 10e + 5 with e ~ {0, 1} a n d u > 5 (so s>_0andife= l a n d s > 1) a n d u = 3 0 s 1 0 e + 1 5 + 10t, w h e r e t is any i n t e g e r such t h a t 3 u / 2 < u _< 4u - 5. I n this case, v >>_3 u / 2 implies t h a t t > - 1 , a n d if t = - 1 t h e n e = 1. T h e case w h e r e t = - 1 a n d e -- 1 is t h e s m a l l e s t e m b e d d i n g just c o n s i d e r e d . So a g a i n w e n e e d only c o n s i d e r t > 0. A l s o , if v > 4u - 5 t h e n v > 3u. A d d t h e set 7/S(s +t+ l) X 7/2 of n e w vertices to V. P a r t i t i o n t h e vertices of V into o n e set of size 4 t + 1 + 2 e a n d 5s - t - 3e + 1 sets of size 4. Since u < 4 u - 5 , w e can a p p l y L e m m a 2.5(3) with x = 5 ( s + t + 1) a n d y = 4 t + 1 + 2e to t h e set o f 4 t + 1 + 2 e vertices. T h e e d g e s in ({s + t + 1},Q,{s + t + 1})x a n d ({2(s + t + 1)},Q,{2(s + t + 1)})x a r e e a c h sets of v e r t e x disjoint 5-cycles. A n d t h e only e d g e s j o i n i n g vertices in W t h a t r e m a i n can, by L e m m a 2.5(3), b e p a r t i t i o n e d into 2 x - 1 - 3y 4 = 10s - 2 t - 6e + 2 1-factors. A s b e f o r e , t h e s e 1-factors can b e u s e d to f o r m 5s - t - 3e + 1 even 2-factors which t o g e t h e r with the 5s - t + 3e + 1 sets of 4 v e r t i c e s o f V a n d using L e m m a 2.2 c o m p l e t e t h e embedding. | W e a r e n o w r e a d y for t h e e x t e n s i o n o f t h e D o y e n - W i l s o n t h e o r e m . THEOREM 3.2. Let (V, P ) be a 5-cycle system of order u. Then (V, P ) can be embedded in a 5-cycle system o f order u > u iff v > 3 u / 2 .
224
NOTE
Proof. T h e r e q u i r e m e n t that v > 3 u / 2 follows directly from L e m m a 1.2. T h e p r o o f of the sufficiency is by i n d u c t i o n o n v. Clearly we c a n a s s u m e t h a t u > 5. By P r o p o s i t i o n (3.1), the e m b e d d i n g is possible if 3 u / 2 < v < 3u. Now let v - 1 or 5 ( m o d 1 0 ) a n d s u p p o s e that for all r - - 1 or 5 (mod 10) with 3 u / 2 < r < v a n d v > 3u, (V, P ) c a n be e m b e d d e d in a 5-cycle system of o r d e r r. Since v >__ 3u + 1, the greatest i n t e g e r less t h a n 2 v / 3 that is c o n g r u e n t to 1 or 5 (mod 10) is certainly at least 2 u - 6. Also, the least i n t e g e r g r e a t e r t h a n 3 u / 2 that is c o n g r u e n t to 1 or 5 ( m o d l 0 ) is at most (3u + 9 ) / 2 (recall that u - 1 or 5 (rood 10)). T h e r e f o r e , if u > 21 t h e n t h e r e exists a n i n t e g e r s -= 1 or 5 (rood 10) with 3 u / 2 < s < 2 v / 3 . Similarly, if u = 15 t h e n v > 51, if u = 11 t h e n v > 35, a n d if u = 5 t h e n v >__ 21, so in every case such a n i n t e g e r s can b e found. I n any case, let t b e the greatest i n t e g e r with 3 u / 2 < t < 2 v / 3 a n d t --- 1 or 5 (rood 10). T h e n by i n d u c t i o n (V, P ) can be e m b e d d e d in a 5-cycle system of o r d e r t, which by P r o p o s i t i o n 3.1 c a n b e e m b e d d e d in a 5-cycle system of o r d e r v. II
ACKNOWLEDGMENTS The second author (C.A.R.) thanks The University of Queensland and the Ethel Raybould Fellowship for support while this research was being done. This research was also supported by the Australian Research Council (DEB) and by the NSF under Grant DMS-9024645 (C.A.R.).
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