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.
Highlights
Snarks are an interesting, important, but somewhat mysterious family of cubic graphs whose characteristic property is that their edges cannot be properly coloured with three colours
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
First we prove that every snark with oddness at least 4, cyclic connectivity 4, and minimum number of vertices can be decomposed into two smaller cyclically 4-edge-connected snarks G1 and G2 by removing a cycle-separating 4-edge-cut, adding at most two vertices to each of the components, and by restoring 3regularity
Summary
Important, but somewhat mysterious family of cubic graphs whose characteristic property is that their edges cannot be properly coloured with three colours. First we prove that every snark with oddness at least 4, cyclic connectivity 4, and minimum number of vertices can be decomposed into two smaller cyclically 4-edge-connected snarks G1 and G2 by removing a cycle-separating 4-edge-cut, adding at most two vertices to each of the components, and by restoring 3regularity. In the second step of the proof we computationally verify that no combination of G1 and G2 can result in a cyclically 4-edgeconnected snark of oddness at least 4 on fewer than 44 vertices This requires checking all suitable pairs of cyclically 4-edge-connected snarks on up to 36 vertices, including those that contain 4-cycles. The smallest currently known cyclically 5-edge-connected snark with oddness at least 4 has 76 vertices (see Steffen [49, Theorem 2.3]), which indicates that a cyclically 5-edgeconnected snark with oddness at least 4 on fewer than 44 vertices either does not exist or will be very difficult to find. We will display a set of 31 such snarks, analyse their properties, and prove that they constitute the complete set of snarks with oddness at least 4 and cyclic connectivity 4 on 44 vertices
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