Research Article  Open Access
Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars
Abstract
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.
1. Introduction
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 [12] 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 [19]. Abueida and Daven [20] investigated the problem of the multidecomposition of the complete graph . Abueida and Daven [21] investigated the problem of the multidecomposition of several graph products where denotes two vertex disjoint edges. Abueida and O'Neil [22] settled the existence problem of the multidecomposition of the complete multigraph for , and . In [23], Priyadharsini and Muthusamy gave necessary and sufficient conditions for the existence of the multidecomposition of where . Furthermore, Shyu [24] investigated the problem of decomposing into paths and stars, and gave a necessary and sufficient condition for . In [25], 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 [26]. Shyu [27] investigated the problem of decomposing into cycles and stars and settled the case . Recently, Lee [28] 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.
2. Preliminaries
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 edgedisjointgraphs. 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 [18]). Let be integers. Then, is decomposable if and only if and
Proposition 2 (see [7]). There exists a decomposition of if and only if , and one of aforementioned (see Table 1) cases occurs.

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 edgedisjoint 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.
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 and be positive integers. Then, has a multidecomposition if and only if and .
Acknowledgment
The authors are grateful to the referees for the valuable comments.
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Copyright © 2013 HungChih Lee and YenPo 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.