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International Journal of Combinatorics
Volume 2013 (2013), Article ID 501701, 6 pages
http://dx.doi.org/10.1155/2013/501701
Research Article

The Linear 2- and 4-Arboricity of Complete Bipartite Graph

1College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
2School of Mathematics, Shandong University, Jinan 250100, China

Received 7 August 2013; Accepted 30 October 2013

Academic Editor: Jun-Ming Xu

Copyright © 2013 Liancui Zuo et al. 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.

Abstract

A linear -forest of an undirected graph is a subgraph of whose components are paths with lengths at most . The linear -arboricity of , denoted by (), is the minimum number of linear -forests needed to decompose . In case the lengths of paths are not restricted, we then have the linear arboricity of , denoted by (). In this paper, the exact value of the linear 2- and 4-arboricity of complete bipartite graph for some and is obtained.

1. Introduction

Throughout this paper, all graphs considered are finite, undirected, and simple. Let represent the set of natural numbers and let denote the set . A graph is -partite (), if it is possible to partition the vertex set into independent sets (called partite sets) such that every edge of joins the vertices in different partite sets. A complete -partite graph is one that is simple and in which each vertex is joined to every vertex that is not in the same subset, which is denoted by if for . For , such graphs are called complete bipartite graphs and denoted by . When , we denote by , which is called balanced complete -partite graph. For , such graphs are called balanced complete bipartite graphs and are denoted by . On the other hand, a graph with order , in which any pair of different vertices are adjacent, is a complete graph, denoted by . Other notations and terminology in the paper are the same as in [1].

A  decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. If a graph has a decomposition , then we say that decompose or can be decomposed into . Furthermore, a linear -forest is a forest whose components are paths of length at most . The linear -arboricity of a graph , denoted by , is the least number of linear -forests needed to decompose .

The notion of linear -arboricity of a graph was first introduced by Habib and Peroche [2]. It is a natural generalization of edge coloring. Clearly, a linear -forest is induced by a matching, and is the chromatic index of a graph . Moreover, the linear -arboricity is also a refinement of the ordinary linear arboricity (or ) which is the case when every component of each forest is a path with no length constraint. By the way, the notion of linear arboricity was introduced earlier by Harary in [3].

In 1982, Habib and Péroche [4] proposed the following conjecture for an upper bound on .

Conjecture 1 (see [4]). If is a graph with maximum degree and , then

For , this is Akiyama’s conjecture [5].

Conjecture 2 (see [5]). Consider  .

So far, there have been a lot of results on the verification of Conjecture 1 in the literature, especially for graphs with particular structures, such as trees [2, 6, 7], regular graphs [8, 9], planar graphs [10], and complete graphs [1115]. The linear arboricity and the linear -arboricity of cubic graphs are studied in recent years, and for more details, please read the papers [11, 1620]. The linear  -arboricity, the linear -arboricity and the low bound of linear -arboricity of balanced complete bipartite graph, and linear 3-arboricity of the balanced complete multipartite graph in [13, 15, 21, 22], respectively, are obtained. In 2010, Xue and Zuo obtained the linear -arboricity of in [23].

As for a low bound on , since any vertex in a linear -forest has degree at most and a linear -forest in a graph has at most edges, the following result is obvious.

Lemma 3 (see [15]). For any graph with maximum degree ,

It is clear that the following lemma holds.

Lemma 4. If , then .

In the following, we will study the linear - and -arboricity of the complete bipartite graph .

2. The Linear 2-Arboricity of Complete Bipartite Graph

Lemma 5. Consider    and .

Proof. By Lemma 3, , , and .
Let , where and . Clearly, can be decomposed into two linear 2-forests: and . Thus .
Let , where and . Clearly, can be decomposed into three linear 2-forests: , , and . Thus .
Let , where and . Clearly, can be decomposed into three linear 2-forests: , , and . Thus .

In the following, we mainly consider the complete bipartite graphs , , and in this section.

Let be a bipartite graph with partite sets and . Suppose that . In [15], Fu et al. defined the bipartite  difference of an edge in as the value . It is not difficult to see that a set consisting of those edges in with the same bipartite difference must be a matching. In particular, such a set is a perfect matching if is a . Furthermore, we can partition the edge set of into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for .

Theorem 6. One has  .

Proof. Let be a complete bipartite graph with partite sets and , where and . Next, we partition and into vertex sets and , respectively, such that and for .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a balanced complete bipartite graph with vertex sets   and . The edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for . Every edge in corresponds to a subgraph of isomorphic to . Since by Lemma 5, each corresponds to two linear 2-forests in . Thus by Lemma 4. Applying Lemma 3, we have . Hence .

Corollary 7. One has  .

Proof. The results can be obtained by Lemma 3 and Theorem 6 immediately.

Theorem 8. , where is odd.

Proof. Let be a complete bipartite graph with partite sets and , where and . Next, we partition into vertex sets and partition into vertex sets , respectively, such that , for , and .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a complete bipartite graph with partite sets and . Let be a balanced complete bipartite graph with partite sets and . Let be a complete bipartite graph with vertex sets   and . In fact, is a star with the center . Thus, . Since the edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for and is odd, the edge set of can be partitioned into pairwise disjoint edge sets such that which consists of a -path and isolated edges , where are taken modulo .
Thus, each edge in corresponds to a subgraph isomorphic to and the 2-path corresponds to a subgraph isomorphic to with partition in . Let where the indices are taken modulo .
After taking away the four edges , and from each to which the 2-path corresponds, each corresponds to two linear 2-forests of by Lemma 5. The edges that we take away form one linear 2-forest. Hence by Lemma 4. Since by Lemma 3, we have .

Corollary 9. , where is odd.

Proof. The results can be obtained by Lemma 3 and Theorem 8 immediately.

Theorem 10. , where is odd.

Proof. The proof is similar to Theorem 8. Let be a complete bipartite graph with partite sets and , where and . Next, we partition into vertex sets and partition into vertex sets , respectively, such that , for , and .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a complete bipartite graph with partite sets and . Let be a balanced complete bipartite graph with vertex sets and . Let be a complete bipartite graph with partite sets and . In fact, is a star with the center . Thus, . Since the edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for and is odd, the edge set of can be partitioned into pairwise disjoint edge sets such that which consists of a -path and isolated edges , where and are taken modulo .
Thus, each edge in corresponds to a subgraph isomorphic to and the 2-path corresponds to a subgraph isomorphic to with partition in . Let where the indices are taken modulo .
After taking away the two edges from each to which the 2-path corresponds, each corresponds to two linear 2-forests of by Lemma 5. The edges that we take away form one linear 2-forest. Hence by Lemma 4. Since by Lemma 3, we obtain .

3. The Linear 4-Arboricity of Complete Bipartite Graph

Lemma 11. One has  .

Proof. By Lemma 3, . In the following, can be partitioned into three linear 4-forests: let , where and . Clearly, can be decomposed into three linear 4-forests , , and (see Figure 1).
Thus .

501701.fig.001
Figure 1: The decomposition of linear 4-forests of .

Lemma 12. .

Proof. By Lemma 3, . In the following, we partition the edge set of into four linear 4-forests: let , where and . Clearly, can be decomposed into four linear 4-forests , , , and (see Figure 2).
Thus .

501701.fig.002
Figure 2: The decomposition of linear 4-forests of .

Lemma 13. .

Proof. By Lemma 3, . In the following, we partition the edge set of into four linear 4-forests: let , where and . can be decomposed into four linear 4-forests , , , and (see Figure 3).
Thus .

501701.fig.003
Figure 3: The decomposition of linear 4-forests of .

In the following, we mainly consider the complete bipartite graphs , , and .

Theorem 14. One has  .

Proof. Let be a complete bipartite graph with partite sets and , where and . Next, we partition and into vertex sets and , respectively, such that and for .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a balanced complete bipartite graph with vertex sets   and . The edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for . Every edge in corresponds to a subgraph of isomorphic to . Since by Lemma 11, each corresponds to three linear 4-forests in . Thus by Lemma 11. Applying Lemma 3, we have
Hence .

Corollary 15. Consider  .

Proof. The results can be obtained by Lemma 3 and Theorem 14 immediately.

Theorem 16. , where is odd.

Proof. Let be a complete bipartite graph with partite sets and , where and . Next, we partition into vertex sets and partition into vertex sets , respectively, such that , for , and .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a complete bipartite graph with partite sets and . Let be a balanced complete bipartite graph with partite sets and . Let be a complete bipartite graph with vertex sets   and . In fact, is a star with the center . Thus, . Since the edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for and is odd, the edge set of can be partitioned into pairwise disjoint edge sets such that which consists of a -path and isolated edges , where are taken modulo .
Thus, each edge in corresponds to a subgraph isomorphic to and the 2-path corresponds to a subgraph isomorphic to with partition in , where is taken modulo . Let where the indices are taken modulo .
After taking away the six edges , and from each to which the 2-path corresponds, each corresponds to three linear 4-forests of by Lemma 12. The edges that we take away form one linear 4-forest (see Figure 2). Hence by Lemma 11. Since by Lemma 3, we have .

Corollary 17. , where is odd.

Proof. The results can be obtained by Lemma 3 and Theorem 16 immediately.

Theorem 18. , where is odd.

Proof. The proof is similar to Theorem 16. Let be a complete bipartite graph with partite sets and , where and . Next, we partition into vertex subsets and partition into vertex subsets , respectively, such that , for , and .
For each , we identify vertices in and denote a new vertex by . For each , we identify vertices in and denote a new vertex by . Then we obtain a complete bipartite graph with partite sets and . Let be a balanced complete bipartite graph with vertex sets   and . Let be a complete bipartite graph with partite sets and . In fact, is a star with the center . Thus, . Since the edge set of can be partitioned into pairwise disjoint perfect matchings such that is exactly the set of edges of bipartite difference in for and is odd, the edge set of can be partitioned into pairwise disjoint edge sets such that which consists of a -path and isolated edges , where are taken modulo .
Thus, each edge in corresponds to a subgraph isomorphic to and the 2-path corresponds to a subgraph isomorphic to with partition in , where is taken modulo . Let where the indices are taken modulo .
After taking away five edges , , , , and from each to which the 2-path corresponds, where the indices are taken modulo , each corresponds to three linear 4-forests of by Lemma 13. The edges that we take away form one linear 4-forest (see Figure 3). Hence by Lemma 11. Since by Lemma 3, we obtain .

Corollary 19. , where is odd.

Proof. The results can be obtained by Lemma 3 and Theorem 18 immediately.

Acknowledgment

This work is supported by NSFC for youth with code of 61103073.

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