ISRNCOMBINATORICS ISRN Combinatorics 2090-8911 Hindawi Publishing Corporation 398473 10.1155/2013/398473 398473 Research Article Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars Lee Hung-Chih Chu Yen-Po da Fonseca C. Feng L. Godbole A. P. Kiliç E. Manstavicius E. Zhou S. Department of Information Technology Ling Tung University Taichung 40852 Taiwan ltu.edu.tw 2013 11 3 2013 2013 30 12 2012 30 01 2013 2013 Copyright © 2013 Hung-Chih Lee and Yen-Po Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let Pk and Sk denote a path and a star with k edges, respectively. For graphs F, G, and H, a (G,H)-multidecomposition of F is a partition of the edge set of F into copies of G and copies of H with at least one copy of G and at least one copy of H. In this paper, necessary and sufficient conditions for the existence of the (Pk, Sk)-multidecomposition of the balanced complete bipartite graph are given.

1. Introduction

Let F, G, and H be graphs. A G-decomposition of F is a partition of the edge set of F into copies of G. If F has a G-decomposition, we say that F is G-decomposable and write GF. A (G,H)-multidecomposition of F is a partition of the edge set of F into copies of G and copies of H with at least one copy of G and at least one copy of H. If F has a (G,H)-multidecomposition, we say that F is (G,H)-multidecomposable and write (G,H)F.

For positive integers m and n, Km,n denotes the complete bipartite graph with parts of sizes m and n. A complete bipartite graph is balanced if m=n. A k-path, denoted by Pk, is a path with k edges. A k-star, denoted by Sk, is the complete bipartite graph K1,k. A k-cycle, denoted by Ck, is a cycle of length k.

P k -decompositions of graphs have been a popular topic of research in graph theory. Articles of interest include . The reader can refer to  for an excellent survey of this topic. Decompositions of graphs into k-stars have also attracted a fair share of interest. Articles of interest include . The study of the (G,H)-multidecomposition was introduced by Abueida and Daven in . Abueida and Daven  investigated the problem of the (Kk,Sk)-multidecomposition of the complete graph Kn. Abueida and Daven  investigated the problem of the (C4,E2)-multidecomposition of several graph products where E2 denotes two vertex disjoint edges. Abueida and O'Neil  settled the existence problem of the (Ck,Sk-1)-multidecomposition of the complete multigraph λKn for k=3,4, and 5. In , Priyadharsini and Muthusamy gave necessary and sufficient conditions for the existence of the (Gn,Hn)-multidecomposition of λKn where Gn,Hn{Cn,Pn-1,Sn-1}. Furthermore, Shyu  investigated the problem of decomposing Kn into k-paths and k-stars, and gave a necessary and sufficient condition for k=3. In , Shyu considered the existence of a decomposition of Kn into k-paths and k-cycles and established a necessary and sufficient condition for k=4. He also gave criteria for the existence of a decomposition of Kn into 3-paths and 3 cycles in . Shyu  investigated the problem of decomposing Kn into k-cycles and k-stars and settled the case k=4. Recently, Lee  established necessary and sufficient conditions for the existence of the (Ck,Sk)-multidecomposition of a complete bipartite graph.

In this paper, we investigate the problem of the (Pk,Sk)-multidecomposition of the balanced complete bipartite graph and give necessary and sufficient conditions for such a multidecomposition to exist.

2. Preliminaries

For our discussions, some terminologies and notations are needed. Let G be a graph. The degree of a vertex x of G, denoted by degGx, is the number of edges incident with x. A graph is r-regular if each vertex is of degree r. The vertex of degree k in Sk is called the center of Sk. Let V and E be subsets of the vertex set and the edge set of G, respectively. We use G[V] to denote the subgraph of G induced by V and G-E to denote the subgraph obtained from G by deleting E. Suppose that G1,G2,,Gt are edge-disjoint-graphs. Then, G1+G2++Gt, or i=1tGi, denotes the graph H with vertex set V(H)=i=1tV(Gi), and edge set E(H)=i=1tE(Gi). Thus, if a graph H can be decomposed into subgraphs G1,G2,,Gt, we write H=G1+G2++Gt, or H=i=1tGi. Moreover, x denotes the smallest integer not less than x and x denotes the largest integer not greater than x. Let v0v1v2,,vk denote the k-path with edges v0v1,v1v2,,vk-1vk, and (v1,v2,,vk) denote the k-cycle with edges v1v2,v2v3,,vk-1vk,vkv1. Throughout the paper, (A,B) denotes the bipartition of Kn,n, where A={a0,a1,,an-1} and B={b0,b1,,bn-1}.

For the edge aibj in Kn,n, the label of aibj is j-i(modn). For example, in K9,9 the labels of a3b7 and a8b2 are 4 and 3, respectively. Note that each vertex of Kn,n is incident with exactly one edge with label i for i=0,1,2,,n-1. Let G be a subgraph of Kn,n and t a nonnegative integer. We use G+t to denote the graph with vertex set {ai:aiV(G)}{bj+t:bjV(G)} and edge set {aibj+t:aibjE(G)}, where the subscripts of b are taken modulo n. In particular, G+0=G.

The following results due to Yamamoto et al. and Parker, respectively, are essential for our discussions.

Proposition 1 (see [<xref ref-type="bibr" rid="B28">18</xref>]).

Let mn1 be integers. Then, Km,n is Sk-decomposable if and only if mk and (1)m0(modk)if  n<k,mn0(modk)if  nk.

Proposition 2 (see [<xref ref-type="bibr" rid="B15">7</xref>]).

There exists a Pk-decomposition of Km,n if and only if mn0(modk), and one of aforementioned (see Table 1) cases occurs.

Case k m n Conditions
1 Even Even Even k 2 m , k2n, both are not equalities
2 Even Even Odd k 2 m - 2 , k2n
3 Even Odd Even k 2 m , k2n-2
4 Odd Even Even k 2 m - 1 , k2n-1
5 Odd Even Odd k 2 m - 1 , kn
6 Odd Odd Even k m , k2n-1
7 Odd Odd Odd k m , kn
3. Main Results

We first give necessary conditions of the (Pk,Sk)-multidecomposition of Kn,n.

Lemma 3.

Let k and n be positive integers. If there exists a (Pk,Sk)-multidecomposition of Kn,n, then kn and n20(modk).

Proof.

The result follows from the fact that the maximum size of a star in Kn,n is n, the size of each member in the multidecomposition is k, and |E(Kn,n)|=n2.

We now show that the necessary conditions are also sufficient. Since Pk=Sk for k=1,2, the result holds for k=1,2 by Proposition 1. So it remains to consider the case k3. The proof is divided into cases n2k, n=k and k<n<2k, which are treated in Lemmas 4, 5, and 6, respectively.

Lemma 4.

Let k and n be positive integers with n2k6. If n20(modk), then Kn,n is (Pk,Sk)-multidecomposable.

Proof.

Let n=qk+r where q and r are integers with 0r<k. Then, q2 from the assumption n2k. Note that (2)Kn,n=Kqk+r,n=K(q-1)k,n+Kk+r,n. By Proposition 1, K(q-1)k,n is Sk-decomposable. On the other hand, trivially, kmin{k+r,n}, and |E(Kk+r,n)|=(k+r)n0(modk) from the assumption n20(modk). This implies that Kk+r,n is Pk-decomposable by Proposition 2. Hence, Kn,n is (Pk,Sk)-multidecomposable.

Lemma 5.

Let k be a positive integer with k3. Then, Kk,k is (Pk,Sk)-multidecomposable.

Proof.

Note that Kk,k=K1,k+Kk-1,k. Trivially, K1,k=Sk. On the other hand, |E(Kk-1,k)|=(k-1)k0(modk). Furthermore, k2(k-1)-1 for odd k, and k2(k-1)=2k-2 for even k. Hence, Kk-1,k is Pk-decomposable by Proposition 2, and Kk,k is (Pk,Sk)-multidecomposable.

Lemma 6.

Let k and n be integers with 3k<n<2k. If n20(modk), then Kn,n is (Pk,Sk)-multidecomposable.

Proof.

Suppose that n=k+r. Then, 0<r<k from the assumption k<n<2k. Let A1={a0,a1,,ak-1}, B1={b0,b1,,bk-1}, A2={ak,ak+1,,ak+r-1}, and B2={bk,bk+1,,bk+r-1}. Let Gi=Kn,n[AiB1] for i=1,2 and H=Kn,n[AB2]. Then, Kn,n=G1+G2+H. Note that G1 is isomorphic to Kk,k, H is isomorphic to Kk+r,r, and G2 is isomorphic to Kr,k, which is Sk-decomposable by Proposition 1. Hence, it is sufficient to show that G1+H is (Pk,Sk)-multidecomposable.

Let t=r2/k. Since kn2, we have kr2, which implies that t is a positive integer Let C=(b0,a0,b1,a1,,bk-2,ak-2,bk-1,ak-1). Then, C is a 2k-cycle in G1. Let p=t/2. For odd t, define a k-path P in G1 as follows: (3)P={b2pa0b2p+1a1b2p-1+k/2ak/2-1b2p+k/2if    k    is    even,b2pa0b2p+1a1b2p+(k-1)/2a(k-1)/2if    k    is    odd, where the subscripts of b are taken modulo k. Since t=r2/k and r<k, we have (4)tr-1k-2. Thus, 2p+1=tk-2, which implies the labels of the edges in P are 2p and 2p+1. Note that for i=0,1,,p-1, C+2i is a 2k-cycle which consists of all of the edges with labels 2i and 2i+1 in G1. Thus, C,C+2,,C+2(p-1) and P are edge-disjoint in G1.

Define a subgraph W of G1 as follows: (5)W={i=0p-1C+2ifor  even  t,i=0p-1C+2i+Pfor  odd    t3,Pfor  t=1. Since C2k can be decomposed into 2 copies of Pk and 2p=t for even t as well as 2p+1=t for odd t, W can be decomposed into t copies of Pk. Let δ=1 for even k and δ=0 for odd k. Note that for even t, degG1-E(W)ai=k-2p=k-t, and for odd t, (6)degG1-E(W)ai={k-t-1if    i=0,1,,k2-2,k-t-δif    i=    k2-1,k-t+1if    i=k2,k2+1,,k-1. Let Xi=(G1-E(W))[{ai}B1] for i=0,1,,k-1. Then for even t, Xi=Sk-t, and for odd t, (7)Xi={Sk-t-1if    i=0,1,,k2-2,Sk-t-δif    i=k2-1,Sk-t+1if    i=k2,k2+1,,k-1 with the center at ai. In the following, we will show that H can be decomposed into r copies of Sk with centers in B2, and into k copies of St with centers in A1 for even t, and into k/2-1 copies of St+1 with centers in {a0,a1,,ak/2-2}, an St+δ with the center at ak/2-1, and k-k/2 copies of St-1 with centers in {ak/2,ak/2+1,,ak-1} for odd t.

We show the required star decomposition of H by orienting the edges of H. For any vertex x of H, the outdegree degH+x (indegree degH-x, resp.) of x in an orientation of H is the number of arcs incident from (to, resp.) x. It is sufficient to show that there exists an orientation of H such that (8)degH+bj=k, where j=k,k+1,,k+r-1, and for even t(9)degH+ai=t, where i=0,1,,k-1, and for odd t(10)degH+ai={t+1if    i=0,1,,k2  -2,t+δif    i=k2-1,t-1if    i=k2,k2+1,,k-1.

We first consider the edges oriented outward from A1 according to the parity of t. Let β=k+(t+1)(k/2-1) and γ=β+t+δ. If t is even, then the edges aibk+ti,aibk+ti+1,,aibk+ti+t-1 are all oriented outward from ai, where i=0,1,,k-1. If t is odd, then the edges aibk+(t+1)i,aibk+(t+1)i+1,,aibk+(t+1)i+t for i=0,1,,k/2-2, and ak/2-1bβ,ak/2-1bβ+1,, ak/2-1bβ+t+δ-1, as well as aib(t-1)(i-k/2)+γ,aib(t-1)(i-k/2)+γ+1,,aib(t-1)(i-k/2)+γ+t-2 for i=k/2,k/2+1,,k-1 are all oriented outward from ai. In both cases, the subscripts of b are taken modulo r in the set of numbers {k,k+1,,k+r-1}. Note that for even t we orient t edges from each ai and for odd t we orient at most t+1 edges from ai. By inequality (4), we have t+1r, which assures us that there are enough edges for the above orientation.

Finally, the edges which are not oriented yet are all oriented from B2 to A. From the construction of the orientation, it is easy to see that (9) and (10) are satisfied, and for all bw,bwB2, we have (11)|degH-bw-degH-bw|1. So, we only need to check (8).

Since degH+bw+degH-bw=k+r for bwB2, it follows from (11) that (12)|degH+bw-degH+bw|1 for bw,bwB2. Note that for even t, i=0k-1degH+ai=tk, and for odd t, (13)i=0k-1degH+ai=(t+1)(k2-1)+t+δ+(t-1)(k-k2)=(t-1)k+2(k2)-1+δ={(t-1)k+2(k2)-1+1if  k  is  even,(t-1)k+2(k+1)2  -1if  k  is  odd.=tk. Thus, (14)w=kk+r-1degH+bw=|E(Kk+r,r)|-i=0k-1degH+ai=(k+r)r-tk=kr+r2-r2=kr. Therefore from (12), we have degH+bw=k for bwB2. This establishes (8). Hence, there exists the required decomposition 𝒟  of H. Let Xi be the star with center at ai in 𝒟 for i=0,1,,k-1. Then, Xi+Xi is a k-star. This completes the proof.

Now, we are ready for the main result. It is obtained form the arguments above, Lemma 4 and Lemmas 3, 4, 5, and 6.

Theorem 7.

Let k and n be positive integers. Then, Kn,n has a (Pk,Sk)-multidecomposition if and only if kn and n20(modk).

Acknowledgment

The authors are grateful to the referees for the valuable comments.

Bouchet A. Fouquet J.-L. Trois types de décompositions d'un graphe en chaînes Annals of Discrete Mathematics 1983 17 131 141 Heinrich K. Liu J. Yu M. P 4 -decompositions of regular graphs Journal of Graph Theory 1999 31 2 135 143 10.1002/(SICI)1097-0118(199906)31:2&lt;135::AID-JGT6&gt;3.3.CO;2-9 MR1686284 ZBL0927.05067 Jacobson M. S. Truszczyński M. Tuza Z. Decompositions of regular bipartite graphs Discrete Mathematics 1991 89 1 17 27 10.1016/0012-365X(91)90396-J MR1108071 ZBL0754.05057 Kotzig A. From the theory of finite regular graphs of degree three and four Časopis pro pěstování matematiky 1957 82 76 92 Kumar C. S. On P4-decomposition of graphs Taiwanese Journal of Mathematics 2003 7 4 657 664 MR2017918 Lee H.-C. Lee M.-J. Lin C. Isomorphic path decompositions of λKn,n,n(λKn,n,nx) for odd n Taiwanese Journal of Mathematics 2009 13 2A 393 402 MR2499995 ZBL1175.05070 Parker C. A. Complete bipartite graph path decompositions [Ph.D. thesis] 1998 Auburn, Ala, USA Auburn University Shyu T.-W. Path decompositions of λKn,n Ars Combinatoria 2007 85 211 219 MR2359292 ZBL1224.05264 Shyu T.-W. Lin C. Isomorphic path decompositions of crowns Ars Combinatoria 2003 67 97 103 MR1973229 ZBL1071.05563 Tarsi M. Decomposition of a complete multigraph into simple paths: nonbalanced handcuffed designs Journal of Combinatorial Theory A 1983 34 1 60 70 10.1016/0097-3165(83)90040-7 MR685212 ZBL0511.05024 Truszczyński M. Note on the decomposition of λKm,n(λKm,n*) into paths Discrete Mathematics 1985 55 1 89 96 10.1016/S0012-365X(85)80023-6 MR793633 ZBL0578.05054 Heinrich K. Path-decompositions Le Matematiche 1992 47 2 241 258 MR1275858 ZBL0790.05065 Bryant D. E. El-Zanati S. Eynden C. V. Hoffman D. G. Star decompositions of cubes Graphs and Combinatorics 2001 17 1 55 59 10.1007/s003730170054 MR1828627 ZBL0984.05069 Lin C. Lin J.-J. Shyu T.-W. Isomorphic star decompositions of multicrowns and the power of cycles Ars Combinatoria 1999 53 249 256 MR1724509 ZBL0994.05113 Tarsi M. Decomposition of complete multigraphs into stars Discrete Mathematics 1979 26 3 273 278 10.1016/0012-365X(79)90034-7 MR535255 ZBL0421.05016 Tazawa S. Decomposition of a complete multipartite graph into isomorphic claws Society for Industrial and Applied Mathematics 1985 6 3 413 417 10.1137/0606043 MR791171 Ushio K. Tazawa S. Yamamoto S. On claw-decomposition of a complete multipartite graph Hiroshima Mathematical Journal 1978 8 1 207 210 MR485773 Yamamoto S. Ikeda H. Shige-eda S. Ushio K. Hamada N. On claw-decomposition of complete graphs and complete bigraphs Hiroshima Mathematical Journal 1975 5 33 42 MR0379300 ZBL0297.05143 Abueida A. Daven M. Multidesigns for graph-pairs of order 4 and 5 Graphs and Combinatorics 2003 19 4 433 447 10.1007/s00373-003-0530-3 MR2030999 ZBL1032.05105 Abueida A. Daven M. Multidecompositions of the complete graph Ars Combinatoria 2004 72 17 22 MR2069042 ZBL1071.05059 Abueida A. Daven M. Multidecompositions of several graph products Graphs and Combinatorics 2012 10.1007/s00373-011-1127-x Abueida A. O'Neil T. Multidecomposition of λKm into small cycles and claws Bulletin of the Institute of Combinatorics and its Applications 2007 49 32 40 MR2285521 ZBL1112.05084 Priyadharsini H. M. Muthusamy A. ( G m , H m ) -multifactorization of λKm Journal of Combinatorial Mathematics and Combinatorial Computing 2009 69 145 150 MR2517315 ZBL1195.05061 Shyu T.-W. Decomposition of complete graphs into paths and stars Discrete Mathematics 2010 310 15-16 2164 2169 10.1016/j.disc.2010.04.009 MR2651813 ZBL1219.05146 Shyu T.-W. Decompositions of complete graphs into paths and cycles Ars Combinatoria 2010 97 257 270 MR2743736 ZBL1249.05313 Shyu T. W. Decomposition of complete graphs into paths of length three and triangles Ars Combinatoria 2012 107 209 224 Shyu T.-W. Decomposition of complete graphs into cycles and stars Graphs and Combinatorics 2011 10. 1007/s00373-011-1105-3 Lee H.-C. Multidecompositions of complete bipartite graphs into cycles and stars Ars Combinatoria 2013 108 355 364