On Galvin orientations of line graphs and list-edge-colouring

TL;DR

This paper introduces and studies Galvin orientations of line graphs, exploring their implications for list-edge-colouring and analyzing their properties in various graph classes.

Contribution

It generalizes Galvin's approach to line graphs, investigates the stronger property of proper Galvin orientations, and examines their presence in specific graph classes.

Findings

Proper Galvin orientations imply list-edge-colourability.

Not all graphs with list-edge-colourability have proper Galvin orientations.

Certain classes like bipartite plus an edge, Petersen graph, and cliques are analyzed.

Abstract

The notion of a Galvin orientation of a line graph is introduced, generalizing the idea used by Galvin in his landmark proof of the list-edge-colouring conjecture for bipartite graphs. If L(G) has a proper Galvin orientation with respect to k, then it immediately implies that G is k-list-edge-colourable, but the converse is not true. The stronger property is studied in graphs of the form `bipartite plus an edge', the Petersen graph, cliques, and simple graphs without odd cycles of length 5 or longer.