Decomposition of a Complete Bipartite Multigraph into Arbitrary Cycle Sizes

Abstract

In a graph G, let $\mu _G(xy)$ denote the number of edges between x and y in G. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and

$$\begin{aligned} \mu _G{(xy)}={\left\{ \begin{array}{ll} \lambda &{}\text{ if }\,\, x\in U \text{ and }\,\, y\in V \text{ or } \text{ if }\,\, x\in V \text{ and }\,\, y\in U\\ 0 &{}\text{ otherwise. } \\ \end{array}\right. } \end{aligned}$$

Let M be the sequence of non-negative integers $m_1,m_2,\ldots ,m_n$. An (M)-cycle decomposition of a graph G is a partition of the edge set into cycles of lengths $m_1,m_2,\ldots ,m_n$. In this paper, we establish some necessary and some sufficient conditions for the existence of an (M)-cycle decomposition of $\lambda K_{v,u}$.