Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars
Let and denote a path and a star with edges, respectively. For graphs , , and , a -multidecomposition of is a partition of the edge set of into copies of and copies of with at least one copy of and at least one copy of . In this paper, necessary and sufficient conditions for the existence of the (, )-multidecomposition of the balanced complete bipartite graph are given.
Let , , and be graphs. A -decomposition of is a partition of the edge set of into copies of . If has a -decomposition, we say that is -decomposable and write . A -multidecomposition of is a partition of the edge set of into copies of and copies of with at least one copy of and at least one copy of . If has a -multidecomposition, we say that is -multidecomposable and write .
For positive integers and , denotes the complete bipartite graph with parts of sizes and . A complete bipartite graph is balanced if . A -path, denoted by , is a path with edges. A -star, denoted by , is the complete bipartite graph . A -cycle, denoted by , is a cycle of length .
-decompositions of graphs have been a popular topic of research in graph theory. Articles of interest include [1–11]. The reader can refer to  for an excellent survey of this topic. Decompositions of graphs into -stars have also attracted a fair share of interest. Articles of interest include [13–18]. The study of the -multidecomposition was introduced by Abueida and Daven in . Abueida and Daven  investigated the problem of the -multidecomposition of the complete graph . Abueida and Daven  investigated the problem of the -multidecomposition of several graph products where denotes two vertex disjoint edges. Abueida and O'Neil  settled the existence problem of the -multidecomposition of the complete multigraph for , and . In , Priyadharsini and Muthusamy gave necessary and sufficient conditions for the existence of the -multidecomposition of where . Furthermore, Shyu  investigated the problem of decomposing into -paths and -stars, and gave a necessary and sufficient condition for . In , Shyu considered the existence of a decomposition of into -paths and -cycles and established a necessary and sufficient condition for . He also gave criteria for the existence of a decomposition of into -paths and cycles in . Shyu  investigated the problem of decomposing into -cycles and -stars and settled the case . Recently, Lee  established necessary and sufficient conditions for the existence of the -multidecomposition of a complete bipartite graph.
In this paper, we investigate the problem of the -multidecomposition of the balanced complete bipartite graph and give necessary and sufficient conditions for such a multidecomposition to exist.
For our discussions, some terminologies and notations are needed. Let be a graph. The degree of a vertex of , denoted by , is the number of edges incident with . A graph is -regular if each vertex is of degree . The vertex of degree in is called the center of . Let and be subsets of the vertex set and the edge set of , respectively. We use to denote the subgraph of induced by and to denote the subgraph obtained from by deleting . Suppose that are edge-disjoint-graphs. Then, , or , denotes the graph with vertex set , and edge set . Thus, if a graph can be decomposed into subgraphs , we write , or . Moreover, denotes the smallest integer not less than and denotes the largest integer not greater than . Let denote the -path with edges , and denote the -cycle with edges . Throughout the paper, denotes the bipartition of , where and .
For the edge in , the label of is . For example, in the labels of and are and , respectively. Note that each vertex of is incident with exactly one edge with label for . Let be a subgraph of and a nonnegative integer. We use to denote the graph with vertex set and edge set , where the subscripts of are taken modulo . In particular, .
The following results due to Yamamoto et al. and Parker, respectively, are essential for our discussions.
Proposition 1 (see ). Let be integers. Then, is -decomposable if and only if and
3. Main Results
We first give necessary conditions of the -multidecomposition of .
Lemma 3. Let and be positive integers. If there exists a -multidecomposition of , then and .
Proof. The result follows from the fact that the maximum size of a star in is , the size of each member in the multidecomposition is , and .
We now show that the necessary conditions are also sufficient. Since for , the result holds for by Proposition 1. So it remains to consider the case . The proof is divided into cases , and , which are treated in Lemmas 4, 5, and 6, respectively.
Lemma 4. Let and be positive integers with . If , then is -multidecomposable.
Proof. Let where and are integers with . Then, from the assumption . Note that By Proposition 1, is -decomposable. On the other hand, trivially, , and from the assumption . This implies that is -decomposable by Proposition 2. Hence, is -multidecomposable.
Lemma 5. Let be a positive integer with . Then, is -multidecomposable.
Proof. Note that . Trivially, . On the other hand, . Furthermore, for odd , and for even . Hence, is -decomposable by Proposition 2, and is -multidecomposable.
Lemma 6. Let and be integers with . If , then is -multidecomposable.
Proof. Suppose that . Then, from the assumption . Let , , , and . Let for and . Then, . Note that is isomorphic to , is isomorphic to , and is isomorphic to , which is -decomposable by Proposition 1. Hence, it is sufficient to show that is -multidecomposable.
Let . Since , we have , which implies that is a positive integer Let . Then, is a -cycle in . Let . For odd , define a -path in as follows: where the subscripts of are taken modulo . Since and , we have Thus, , which implies the labels of the edges in are and . Note that for , is a -cycle which consists of all of the edges with labels and in . Thus, and are edge-disjoint in .
Define a subgraph of as follows: Since can be decomposed into copies of and for even as well as for odd , can be decomposed into copies of . Let for even and for odd . Note that for even , , and for odd , Let for . Then for even , , and for odd , with the center at . In the following, we will show that can be decomposed into copies of with centers in , and into copies of with centers in for even , and into copies of with centers in , an with the center at , and copies of with centers in for odd .
We show the required star decomposition of by orienting the edges of . For any vertex of , the outdegree (indegree , resp.) of in an orientation of is the number of arcs incident from (to, resp.) . It is sufficient to show that there exists an orientation of such that where , and for even where , and for odd
We first consider the edges oriented outward from according to the parity of . Let and . If is even, then the edges are all oriented outward from , where . If is odd, then the edges for , and , , as well as for are all oriented outward from . In both cases, the subscripts of are taken modulo in the set of numbers . Note that for even we orient edges from each and for odd we orient at most edges from . By inequality (4), we have , which assures us that there are enough edges for the above orientation.
Finally, the edges which are not oriented yet are all oriented from to . From the construction of the orientation, it is easy to see that (9) and (10) are satisfied, and for all , we have So, we only need to check (8).
Since for , it follows from (11) that for . Note that for even , , and for odd , Thus, Therefore from (12), we have for . This establishes (8). Hence, there exists the required decomposition of . Let be the star with center at in for . Then, is a -star. This completes the proof.
Theorem 7. Let and be positive integers. Then, has a -multidecomposition if and only if and .
The authors are grateful to the referees for the valuable comments.
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