Edge Cover Colouring Versus Minimum Degree in Multigraphs

TL;DR

This paper investigates the maximum number of parts in an edge cover colouring of multigraphs, establishing tight bounds related to the minimum degree and extending classical edge-colouring results.

Contribution

It provides a tight lower bound on the number of parts in edge cover colourings as a function of minimum degree, extending Shannon's theorem to this new context.

Findings

Lower bound on edge cover colouring parts is tight except for specific degrees.

Bounds differ by one in certain cases, indicating near-optimal results.

Extends classical edge-colouring theory to edge cover colourings in multigraphs.

Abstract

An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum degree delta, which would be an analogue of Shannon's theorem from 1949 on edge-colouring multigraphs. We are able to give a lower bound that is tight except when delta=9 or delta is odd and > 12; in these non-tight cases the best upper and lower bounds differ by one.