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