On measures of edge-uncolorability of cubic graphs: A brief survey and some new results

TL;DR

This paper surveys measures of how far cubic graphs, especially snarks, are from being 3-edge-colorable, explores their relationships, and presents new results that contribute to understanding key conjectures in graph theory.

Contribution

It introduces new results on measures of edge-uncolorability in cubic graphs and analyzes their relationships, advancing understanding of snarks and related conjectures.

Findings

New bounds on edge-uncolorability measures

Relationships between different parameters studied

Partial progress towards key conjectures

Abstract

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called {\em snarks}, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.