c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. DISCRETE MATH. Vol. 20, No. 3, pp c 2006 Society for Industrial Applied Mathematics CYCLE DECOMPOSITIONS OF K n,n I JUN MA, LIQUN PU, AND HAO SHEN Abstract. Let K n,n denote the complete bipartite raph with n vertices in each bipartition set K n,n I denote K n,n with a -factor removed. An m-cycle system of K n,n I is a collection T of m-cycles such that each ede of K n,n I is contained in a unique m-cycle of T. In this paper, it is proved that the necessary sufficient conditions for the existence of an m-cycle system of K n,n I are n (mod 2), m 0 (mod 2), 4 m 2n, n(n ) 0(modm). Key words. decomposition, cycle, complete bipartite raph, -factor AMS subject classification. 05C38 DOI. 0.37/ Introduction. Let G be a raph with vertex set V (G) ede set E(G). An m-cycle system of G is a collection T of m-cycles such that each ede of G is contained in a unique m-cycle of T. It is natural to ask when there exists an m-cycle system of G. It is not difficult to verify that the followin conditions are necessary for the existence of an m-cycle system of G: 3 m V (G), E(G) 0 (mod m), d(u) 0 (mod 2) for each u V (G), where d(u) denotes the number of edes incident with u in G. Let K v denote a complete raph of order v, K v I denote K v with a -factor removed, K x,y denote a complete bipartite raph with partite sets of sizes x y. When G is K v, K v I, ork x,y, the existence problem of m-cycle systems of G has been completely settled [2, 3, 4]. Theorem (see [3, 4]). Let m v be positive inteers. Then there exists an m-cycle system of K v if only if v (mod 2), 3 m v, v(v ) 0 (mod 2m). Theorem 2 (see [3, 4]). Let m v be positive inteers. Then there exists an m-cycle system of K v I if only if v 0 (mod 2), 3 m v, v(v 2) 0 (mod 2m). Theorem 3 (see [2]). Let m 0 (mod 2) m 4. Then there exists an m-cycle system of K x,y if only if x, y 2m, x y 0 (mod 2), xy 0 (mod m). In this paper, we consider the case when G is K x,y I, where K x,y I denotes K x,y with a -factor removed. Obviously if K x,y has a -factor, then necessarily we have x = y. Let x = y = n. Simple countin ives the followin necessary conditions. Received by the editors March 9, 2005; accepted for publication (in revised form) February 23, 2006; published electronically Auust 25, This project was supported by National Natural Science Foundation of China under rant Department of Mathematics, Shanhai Jiao Ton University, Shanhai , People s Republic of China (mj904@sjtu.edu.cn, sarah 009@sjtu.edu.cn, haoshen@sjtu.edu.cn). 603

2 604 JUN MA, LIQUN PU, AND HAO SHEN Lemma 4. If there exists an m-cycle system of K n,n I, then n (mod 2), m 0 (mod 2), 4 m 2n, n(n ) 0 (mod m). It was proved [5] that m-cycle systems of K n,n I exist in the special case m =2n. The followin theorem was obtained by Archdeacon et al. []. Theorem 5. Let m 0 (mod 2), m 4, n (mod 2). Ifm 0 (mod 4) m<n,orifm 2 (mod 4) m<2n, then there exists an m-cycle system of K n,n I if only if n(n ) 0 (mod m). But, it is still not known whether m-cycle systems of K n,n I exist when n (mod 2), n 3, m 0 (mod 4), n<m<2n, n(n ) 0 (mod m). The main purpose of this paper is to determine the existence of m-cycle systems of K n,n I for the open case in Theorem 5. In fact, we will ive a unified simple proof to the followin theorem. Theorem 6. Let m n be positive inteers. Then there exists an m-cycle system of K n,n I if only if n (mod 2), m 0 (mod 2) 4 m 2n, n(n ) 0 (mod m). 2. Cycle decomposition of K n,n I with 2 m n 3 m. A cycle on m 2 vertices is denoted by C m. If a raph G is the ede-disjoint union of m-cycles, then we say that G is C m -decomposable. We shall also write C m G. In [], Archdeacon et al. skillfully proved the followin lemma. Lemma 7. Let m 2(mod 4), n (mod 2), 6 m 2n. Then C m K n,n I if only if m n(n ). By Lemma 7, we obtain the followin corollary immediately. Corollary 8. Let m 2 (mod 4) m 6. If n 2 m, 3 2 m}, then C m K n,n I. Now, for a positive inteer n, let D Z n X(n; D) be a raph with vertex set Z n Z 2 ede set i 0, (i + d) } d D, i Z n }. Clearly, K n,n can be viewed as X(n; Z n ). The elements of D are called (0,)-mixed differences. We say that i 0, (i + d) } is an ede of difference d. Suppose that C =((i ) 0, (i 2 ),...,(i m ) 0, (i m ) )isac m in X(n; D). For x Z n, let C + x =((i + x) 0, (i 2 + x),...,(i m + x) 0, (i m + x) ). Obviously, C + x is still a C m. Let (C) =C + x x Z n }. Here, (C) is called the orbit enerated by C, C is called a base cycle of (C). For any inteer x, let ε(x) = 0 if x 0 (mod 2), if x (mod 2). We use the difference method to ive constructions of m-cycle systems of X(n; D) which we need in this paper. Lemma 9. Let m be a positive inteer. If m 0 (mod 2) m 4, then C m K m+,m+ I. Proof. We view K m+,m+ as X(m +;Z m+ ). For r = 0,,...,m, define

3 CYCLE DECOMPOSITIONS OF K n,n I 605 d r Z m+ as 0 if r =0, ε( 2m) if r =, d r = r if 2 r 2 m ε( 2 m), r + if 2 m + ε( 2m) r m, 2 m + ε( 2m) if r = m. Let e k = k r=0 ( )r d r for 0 k m. Let 0 if k< 2 θ = m + ε( 2 m), if k 2 m + ε( 2 m). Then 0 if k =0orm, e k = 2 k + ε( 2m) if k 0 (mod 2) 2 k m 2, 2 (k +) θ + ε( 2m) if k (mod 2). Let C be the followin closed trail: (e 0 ), (e ) 0, (e 2 ), (e 3 ) 0,...,(e m 2 ), (e m ) 0, (e m ). The differences used in C are d,d 2,...,d m. 0=e 0 <e 2 <e 4 < <e m 2 = ( ) 2 m +ε 2 m m + ε ( ) 2 m = e + m +>e 3 + m +>e 5 + m +> >e m + m + = ( ) 2 m + ε 2 m, then C is an m-cycle. Let T =(C) I = (i) 0, (i + ε( 2 m)) } i Z m+ }. Then I is a -factor in K m+,m+, T is an m-cycle system of K m+,m+ I, C m K m+,m+ I. Lemma 0. Let u be an inteer let m 0 (mod 2), m 4, n (mod 2), m<2n, = cd(m, n) >, h m.forr =, 2,..., m, define d r Z n as u m + r if r h, d r = u m + r + if h r m, u m + m 2 + ε(h)+ n if r = m. Let D = d,d 2,...,dm }. Then C m X(n; D). Proof. Let d 0 = 0 e k = k r=0 ( )r d r for 0 k m. Let θ = 0 if k<h, if k h.

4 606 JUN MA, LIQUN PU, AND HAO SHEN Then 2 k + θ( ε(h)) if k 0 (mod 2) 0 k m 2, e k = u m 2 (k +) θε(h) if k (mod 2), n if k = m. Let P be the followin trail: (e 0 ), (e ) 0, (e 2 ), (e 3 ) 0,...,(em 2 ), (e m ) 0, (e m ). The differences used in P are d,d 2,...,dm. 0=e 0 <e 2 <e 4 < <em 2 = m 2 ε(h) u m θε(h) =e >e 3 >e 5 > >em = u m m 2 θε(h), P is a path. Moreover, the first last vertices are the only ones which are conruent modulo n. It follows that C = P ( P + n ) ( P + 2n ) ( P + ) ( )n is an m-cycle. In C, each difference in D occurs exactly times, for each j Z 2, if vertices (i ) j (i 2 ) j are both incident with edes of difference d, then i i 2 (mod n ). Let T =(C). It follows that T is an m-cycle system of X(n; D) C m X(n; D). Lemma. Let m 0 (mod 2), m 4, n (mod 2), m<2n, = cd(m, n) >, h (mod 2), 2 h m.forr =, 2,..., m, define d r Z n as 0 if r =, r if 2 r<h, d r = r + if h r m, m 2 + n if i = m. Let D = d,d 2,...,dm }. Then C m X(n; D). Proof. Let d 0 = 0 e k = k r=0 ( )r d r for k m. Furthermore, let 0 if k<h, θ = if k h. Then e k = 0 if k =0, 2 k + if k 0 (mod 2) 2 k m 2, 2 (k ) θ if k (mod 2), n if k = m. Let P be the followin trail: (e 0 ), (e ) 0, (e 2 ), (e 3 ) 0,...,(e m 2 ), (e m ) 0, (e m ).

5 The differences used in P are d,d 2,...,dm. CYCLE DECOMPOSITIONS OF K n,n I 607 0=e 0 <e 2 <e 4 < <em 2 = m 2 n 0=e >e 3 >e 5 > >em = m 2 + θ ( n ), P is a path. Moreover, the first last vertices are the only ones which are conruent modulo n. It follows that C = P ( P + n ) ( P + 2n ) ( P + ) ( )n is an m-cycle. In C, each difference in D occurs exactly times, for each j Z 2, if vertices (i ) j (i 2 ) j are both incident with edes of difference d, then i i 2 (mod n ). Let T =(C). It follows that T is an m-cycle system of X(n; D) C m X(n; D). With the above preparations, we now prove the followin theorem. Theorem 2. Let m n be positive inteers. If m 0 (mod 2), m 4, n (mod 2), 2 m<n< 3 2 m, n(n ) 0 (mod m), then C m K n,n I. Proof. When = cd(m, n) =,n = m + since n(n ) 0 (mod m). So, C m K m+, m+ I by Lemma 9. If n m+, then >. n(n ) 0 (mod m), we have n 0 (mod m ). Let s = (n ) m. We view K n,n as X(n; Z n ). For t =0,,...,s, let D t = 0,,..., m } if t =0, t m +,tm +2,...,tm + m } if t s. Let δ = ε(s)[ ε( 2 m)]. For t =0,,...,s, define h t as t m h t = + m 2 + ε(t)[ ε( 2 m)] + n if 0 t s 2, n m 2 +δ ε( 2m) if t = s. Observe that h t D t+ for 0 t s 2, h s D 0, h s δ. Thus, we let where d D 0 \δ} ε(d) =δ. For t s, let ˆD 0 = (D0 h 0 })\h s,δ} if h s >δ, (D 0 h 0 })\h s,d} if h s = δ, ˆD t =(D t \h t }) h t }. When t =0,h 0 = m 2 + n. δ = 0 or, there are the followin two cases. Case. δ =0.

6 608 JUN MA, LIQUN PU, AND HAO SHEN Then h s = n m 2 ε( 2 m). It is easy to check that ε(h s ) = 0. We take u = 0 hs if h h = s > 0, d if h s =0. Clearly, h 0 = m 2 + n + ε(h). By Lemma 0, we have C m X(n; ˆD 0 ). Case 2. δ =. Then h s = n m 2 ε( 2 m). It is easy to check that ε(h s ) =. We take u = 0 hs if h h = s >, d if h s =. Clearly, h 2. By Lemma, we have C m X(n; ˆD 0 ). For each t =, 2,...,s, we take u = t h = h t t m. It is easy to check that h t = u m + m 2 + ε(h)+ n. By Lemma 0, we have C m X(n; ˆD t ). Clearly, s t=0 ˆD t = Zn \δ} if h s >δ, Z n \d} if h s = δ, ˆD t ˆD r = φ for t r. Suppose that T t is an m-cycle system of X(n; ˆD t ) for 0 t s. Let T = s t=0 T t i0, (i + δ) I = } i Z n } if h s >δ, i 0, (i + d) } i Z n } if h s = δ. Then T is an m-cycle system of K n,n I C m K n,n I. 3. The proof of Theorem 6. Now, we are in a position to prove the main theorem of this paper. Proof of Theorem 6. For 2 m n 3 2 m, we have C m K n,n I by Corollary 8 Theorem 2. If n> 3 2 m, then we may write n = qm + r with 2 m<r 3 2m q. we have n (mod 2) n(n ) 0 (mod m), r (mod 2) r(r ) 0 (mod m). Suppose the vertex set of K n,n is v 0,v,...,v qm+r } u 0,u,...,u qm+r } I = u i,v i } 0 i qm + r } is a -factor in K n,n. For i q, let V i = v (i )m+j j m} U i = u (i )m+j j m}. Let V q+ = v qm+j j r } U q+ = u qm+j j r }.

7 CYCLE DECOMPOSITIONS OF K n,n I 609 For i q +, let H i,i be the subraph of K n,n I induced by (V i v 0 }) (U i u 0 }). By Corollary 8 Theorem 2, C m H i,i. Let T i,i be an m-cycle system of H i,i. For i, j q + i j, let H i,j be the subraph of K n,n I induced by V i U j. By Theorem 3, C m H i,j. Let T i,j be an m-cycle system of H i,j. Let T = i,j q+ T i,j. Then T is an m-cycle system of K n,n I C m K n,n I. This completes the proof. Acknowledment. The authors are thankful to the referees for their helpful comments to improve the paper. REFERENCES [] D. Archdeacon, M. Debowsky, J. Dinitz, H. Gavlas, Cycle systems in the complete bipartite raph minus a one-factor, Discrete Math., 284 (2004), pp [2] D. Sotteau, Decompositions of K m,n(km,n ) into cycles (circuits) of lenth 2k, J. Combin. Theory Ser. B, 29 (98), pp [3] B. Alspach H. Gavlas, Cycle decompositions of K n K n I, J. Combin. Theory Ser. B, 8 (200), pp [4] M. Šajna, Cycle decompositions, III: Complete raphs fixed lenth cycles, J. Combin. Des., 0 (2002), pp [5] R. Laskar B. Auerbach, On decomposition of r-partite raphs into ede-disjoint Hamilton circuits, Discrete Math., 4 (976), pp