Packing paths in a $(\lambda+\mu)K_{v+u}-\lambda K_v$

TL;DR

This paper establishes necessary and sufficient conditions for decomposing a specific multigraph, constructed from complete graphs with multiple edges, into paths of fixed length, advancing understanding of graph path packings.

Contribution

It provides the first complete characterization of when such multigraphs can be decomposed into paths of a given length.

Findings

Derived necessary and sufficient conditions for path decompositions.

Extended existing graph decomposition theories to multigraphs with multiple edges.

Solved a specific open problem in graph packing and decomposition.

Abstract

Following standard terminology, $\lambda K_v$ is a multigraph on $v$ vertices such that $\lambda$ edges join each pair of vertices. Let $(\lambda+\mu)K_{v+u}-\lambda K_v$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and $(\lambda+\mu)-\lambda=\mu$ edges between the vertices $x$ and $y$ if $x$ and $y$ both lie in $V$ and $\lambda+\mu$ edges between $x$ and $y$ otherwise. The main result of this paper establishes necessary and sufficient conditions for an $m$-path decomposition of $(\lambda+\mu)K_{v+u}-\lambda K_v$.