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

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.

The smallest nontrivial snarks of oddness 4

“…This section collects the most basic definitions and notation needed for understanding the present paper. For a more detailed introduction to the topic we refer the reader to our preceding paper [8].…”

“…By the Parity Lemma, every colouring of a 4-pole has one of the following types: 1111, 1122, 1212, and 1221 (for a precise definition of the type of a colouring see [8]). Observe that every colourable 4-pole admits at least two different types of colourings.…”

“…Let S 36 denote the set of all cyclically 4-edge-connected snarks with at most 36 vertices; as mentioned in [8], the set S 36 consists of 432 105 682 nonisomorphic graphs. For any two snarks G 1 and G 2 from S 36 let us apply the following operation:…”

“…This paper is a sequel of [8] where we have proved that the smallest number of vertices of a snark -a connected cubic graph whose edges cannot be properly coloured with three colours -which has cyclic connectivity 4 and oddness at least 4 is 44. The purpose of the present paper is to show that there are precisely 31 such snarks, all of them having oddness exactly 4, resistance 3, and girth 5.…”

“…The purpose of the present paper is to show that there are precisely 31 such snarks, all of them having oddness exactly 4, resistance 3, and girth 5. Together with [8], this paper provides a partial answer to the following question posed in [4,Problem 2], leaving open the existence of cyclically 5-edge-connected snarks of oddness at least 4 on fewer than 44 vertices:…”