Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

TL;DR

This paper determines that the smallest nontrivial snark with oddness at least 4 and cyclic connectivity 4 has 44 vertices, using structural analysis and extensive computations to extend known classifications.

Contribution

It establishes the exact minimal order of such snarks, extending the classification of cyclically 4-edge-connected snarks to 36 vertices and testing related conjectures.

Findings

Smallest snark with oddness ≥ 4 and cyclic connectivity 4 has 44 vertices.

Extended the classification of cyclically 4-edge-connected snarks up to 36 vertices.

Provided data to test conjectures involving minimal snarks as counterexamples.

Abstract

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.