Critical and flow-critical snarks coincide

TL;DR

This paper proves that two previously distinct notions of criticality in snarks, related to 3-edge-colorings and 4-flows, are actually equivalent, unifying the theory of snark criticality.

Contribution

The paper establishes the equivalence of critical and flow-critical snarks, resolving a long-standing open question and completing prior partial results.

Findings

Critical and flow-critical snarks are the same class.

Completes previous partial results on snark criticality.

Unifies different notions of criticality in snarks.

Abstract

Over the past twenty years, critical and bicritical snarks have been appearing in the literature in various forms and in different contexts. Two main variants of criticality of snarks have been studied: criticality with respect to the non-existence of a $3$-edge-colouring and criticality with respect to the non-existence of a nowhere-zero $4$-flow. In this paper we show that these two kinds of criticality coincide, thereby completing previous partial results of de Freitas et al. [Electron. Notes Discrete Math. 50 (2015), 199--204] and Fiol et al. [ arXiv:1702.07156v1 (2017)].