Weak Snark

A weak snark is a cyclically 4-edge connected cubic graph with 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 are only possible for graphs with even vertex count. The numbers of weak snarks on 2, 4, 6, ... nodes 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 (strong) snarks have 22 nodes and are illustrated above.

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Brinkmann, G.; Goedgebeur, J.; Hägglund, J.; and Markström, K. "Generation and Properties of Snarks." J. Comb. Th. 103, 468-488, 2013.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."

Cite this as:

Weisstein, Eric W. "Weak Snark." From MathWorld--A Wolfram Web Resource.

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