3-critical subgraphs of snarks

TL;DR

This paper explores the structure of snarks by analyzing their minimal conflicting subgraphs, establishing relationships with their edge-colourings, resistance, and critical subgraphs, and introducing the concept of clusters of these subgraphs.

Contribution

It introduces the concept of minimal conflicting subgraphs in snarks and characterizes their relationship with resistance and critical subgraphs, advancing structural understanding.

Findings

Resistance equals the minimum edges covering all minimal conflicting subgraphs.

Characterization of the critical subgraph in relation to minimal conflicting subgraphs.

Introduction of clusters of minimal conflicting subgraphs and their properties.

Abstract

In this paper we further our understanding of the structure of class two cubic graphs, or snarks, as they are commonly known. We do this by investigating their 3-critical subgraphs, or as we will call them, minimal conflicting subgraphs. We consider how the minimal conflicting subgraphs of a snark relate to its possible minimal 4-edge-colourings. We fully characterise the relationship between the resistance of a snark and the set of minimal conflicting subgraphs. That is, we show that the resistance of a snark is equal to the minimum number of edges which can be selected from the snark, such that the selection contains at least one edge from each minimal conflicting subgraph. We similarly characterise the relationship between what we call \textit{the critical subgraph} of a snark and the set of minimal conflicting subgraphs. The critical subgraph being the set of all edges which are conflicting in some minimal colouring of the snark. Further to this, we define groups, or \textit{clusters}, of minimal conflicting subgraphs. We then highlight some interesting properties and problems relating to clusters of minimal conflicting subgraphs.