Hamilton path decompositions of complete multipartite graphs

TL;DR

This paper establishes necessary and sufficient conditions for decomposing complete multipartite graphs into edge-disjoint Hamilton paths, based on the graph's edge count and maximum degree.

Contribution

It provides a complete characterization of when such decompositions into Hamilton paths are possible for complete multipartite graphs.

Findings

Decomposition into Hamilton paths is possible if and only if m/(n-1) is an integer.

The maximum degree of the graph must be at most 2m/(n-1).

The paper offers a precise criterion linking edge count, maximum degree, and Hamilton path decompositions.

Abstract

We prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $\frac m{n-1}$ is an integer and the maximum degree of $K$ is at most $\frac {2m}{n-1}$.