Cubic graphs with colouring defect 3

TL;DR

This paper studies the structure of snarks with a colouring defect of 3, developing reduction techniques and analyzing Fano flows, with implications for longstanding conjectures in graph theory.

Contribution

It introduces a reduction framework for snarks with defect 3 and connects these structures to Fano flows, advancing understanding of their properties.

Findings

Snarks with defect 3 can be reduced to simpler forms or derived from larger snarks.

Every defect 3 snark can be simplified to a nontrivial snark or constructed from one by vertex inflation.

Analysis of Fano flows is key to understanding the structure of defect 3 snarks.

Abstract

The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While $3$-edge-colourable graphs have defect $0$, those that cannot be $3$-edge-coloured (that is, snarks) are known to have defect at least $3$. In this paper we focus on the structure and properties of snarks with defect $3$. For such snarks we develop a theory of reductions similar to standard reductions of short cycles and small cuts in general snarks. We prove that every snark with defect $3$ can be reduced to a snark with defect $3$ which is either nontrivial (cyclically $4$-edge-connected and of girth at least $5$) or to one that arises from a nontrivial snark of defect greater than $3$ by inflating a vertex lying on a suitable $5$-cycle to a triangle. The proofs rely on a detailed analysis of Fano flows associated with triples of perfect matchings leaving exactly three uncovered edges. In the final part of the paper we discuss application of our results to the conjectures of Berge and Fulkerson, which provide the main motivation for our research.