Critical and flow-critical snarks coincide

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 nowherezero 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].

Morphology of Small Snarks

“…Critical snarks that are not bicritical, called strictly critical, appear to be very rare. This can be observed already among snarks of small order: there are exactly 55172 critical snarks of order not exceeding 36, but only 846 of them are strictly critical, just slightly over 1.5 percent [5,34].…”

“…Order 10 20 22 24 26 28 30 32 34 36 38 NN substitution 0 0 2 0 0 0 10 11 26 10 ! 39 TT substitution 0 0 0 0 8 0 0 0 84 69 !…”

“…If we take into account the fact that contracting an edge has the same effect on the existence of a nowhere-zero flow as identifying its end-vertices, 4-flow-edge-critical snarks and 4-flow-vertex-critical snarks are natural counterparts of critical and bicritical snarks, respectively. Nevertheless, it has been only recently shown [34] that, in spite of different formal definitions, flow-critical snarks are the same as critical snarks.…”

Morphology of small snarks

“…Critical snarks that are not bicritical, called strictly critical, appear to be very rare. This can be observed already among snarks of small order: there are exactly 55172 critical snarks of order not exceeding 36, but only 846 of them are strictly critical, just slightly over 1.5 percent [5,33].…”

“…If we take into account the fact that contracting an edge has the same effect on the existence of a nowhere-zero flow as identifying its end-vertices, 4-flow-edge-critical snarks and 4-flow-vertex-critical snarks are natural counterparts of critical and bicritical snarks, respectively. Nevertheless, it has been only recently shown [33] that in spite of different formal definitions flow-critical snarks are exactly the same as critical snarks. Theorem 2.…”