On Snarks that are far from being 3-Edge Colorable

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.

Treelike Snarks

“…As for cubic bridgeless graphs G with χ e (G) ≥ 5, it was asked by Fouquet and Vanherpe [9] whether the Petersen graph was the only such graph that is cyclically 4-edgeconnected. Hägglund [12] constructed another example (of order 34) and asked for a characterisation of such graphs [12,Problem 3]. Esperet and Mazzuoccolo [7] generalized Hägglund's example to an infinite family.…”

Measures of Edge-Uncolorability of Cubic Graphs

“…These two questions were asked by Hägglund [51] for the oddness. Clearly, for the oddness it suffices to consider even numbers.…”

The smallest nontrivial snarks of oddness 4

“…The oddness of a bridgeless cubic graph G is the smallest number of odd circuits in a 2-factor of G, and the resistance of G is the smallest number of vertices (or edges) of G whose removal yields a 3-edge-colourable graph. Both invariants are important measures of uncolourability of cubic graphs and have been investigated by numerous authors [1,5,11,12,13,15,25]. One of the reasons why these invariants have recently received so much attention resides in the fact that snarks with large resistance or oddness may provide potential counterexamples to several profound conjectures such as the cycle double cover conjecture, the 5-flow conjecture, and others [11,13,14].…”