On the Smallest Snarks with Oddness 4 and Connectivity 2

  • Jan Goedgebeur
Keywords: Cubic graph, Snark, Chromatic index, Oddness, Computation, Exhaustive search


A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph.

Lukot'ka, Máčajová, Mazák and Škoviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked that there are exactly two such graphs of that order. However, this remark is incorrect as — using an exhaustive computer search — we show that there are in fact three snarks with oddness 4 on 28 vertices. In this note we present the missing snark and also determine all snarks with oddness 4 up to 34 vertices.
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