On Snarks that are far from being 3-Edge Colorable

Jonas Hägglund

doi:10.37236/3430

Abstract

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

Highlights

A cubic graph is said to be colorable if it has a 3-edge coloring and uncolorable otherwise
It it well known that an edge minimal counterexample to some classical conjectures in graph theory, such as the cycle double cover conjecture [15, 18], Tutte’s 5-flow conjecture [19] and Fulkerson’s conjecture [4], must reside in this family of graphs
The oddness of a bridgeless cubic graph G is defined as the minimum number of odd components in any 2-factor in G and is denoted by o(G)

Summary

Introduction

A cubic graph is said to be colorable if it has a 3-edge coloring and uncolorable otherwise. It it well known that an edge minimal counterexample (if such exists) to some classical conjectures in graph theory, such as the cycle double cover conjecture [15, 18], Tutte’s 5-flow conjecture [19] and Fulkerson’s conjecture [4], must reside in this family of graphs. The oddness of a bridgeless cubic graph G is defined as the minimum number of odd components in any 2-factor in G and is denoted by o(G) Another measure is the resistance of G, r3(G), which was introduced by Steffen [17] and is defined as the minimal number of edges that needs to be removed from G in order to obtain a 3-edge colorable graph, i.e. r3(G) := min{|M | : M ⊂ E(G) and χ (G − M ) = 3}. We give simple constructions for two infinite families of snarks with high oddness and resistance compared to the number of vertices

The construction

Perfect matching covers

Cycle double covers

Circumference

Concluding remarks