Weak Snark -- from Wolfram MathWorld

A weak snark is a finite simple graph that is connected and cubic, with cyclic edge connectivity at least 4, edge chromatic number 4, and girth at least 4 (Brinkmann et al. 2013). Weak snarks therefore represent a more general class than (and include) the usual snarks (which must have girth at least 5).

Like snarks, weak snarks have even graph order. The numbers of nonisomorphic weak snarks on 2, 4, 6, ... vertices are 0, 0, 0, 0, 1, 0, 0, 0, 2, 6, 31, 155, 1297, 12517, 139854, 1764950, 25286953, 404899916, ... (OEIS A216834).

The smallest weak snarks that are not also snarks have 22 vertices and are illustrated above.

References

Brinkmann, G.; Goedgebeur, J.; Hägglund, J.; and Markström, K. "Generation and Properties of Snarks." J. Combin. Th. Ser. B 103, 468-488, 2013. https://doi.org/10.1016/j.jctb.2013.05.001.

Hägglund, J. and Markström, K. "On Stable Cycles and Cycle Double Covers of Graphs with Large Circumference." Disc. Math. 312, 2540-2544, 2012.

Holton, D. A. and Sheehan, J. "Snarks." Ch. 3 in The Petersen Graph. Cambridge, England: Cambridge University Press, pp. 79-111, 1993.

Sloane, N. J. A. Sequence A216834 in "The On-Line Encyclopedia of Integer Sequences."

Weak Snark

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