Decomposition of a complete bipartite multigraph into arbitrary cycle sizes

TL;DR

This paper establishes the precise conditions under which a complete bipartite multigraph can be decomposed into cycles of specified lengths, generalizing previous results to multigraphs with arbitrary cycle sizes.

Contribution

It provides necessary and sufficient conditions for $(M)$-cycle decompositions of complete bipartite multigraphs, extending classical cycle decomposition results to multigraphs with arbitrary cycle lengths.

Findings

Derived conditions for cycle decompositions in bipartite multigraphs.

Extended classical cycle decomposition results to multigraphs.

Characterized when such decompositions exist based on graph parameters.

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 \[ \mu_G(xy)=\begin{cases} \lambda &\mbox{if $x\in U$ and $y\in V$ or if $x\in V$ and $y\in U$}\\ 0 &\mbox{otherwise.} \\ \end{cases} \] Let $M$ be a 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 necessary and sufficient conditions for the existence of an $(M)$-cycle decomposition of $\lambda K_{v,u}$.