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arXiv · 1710.00757

On the smallest snarks with oddness 4 and connectivity 2

Jan Goedgebeur
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Abstract

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, M\'acajov\'a, Maz\'ak and \v{S}koviera 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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Jan Goedgebeur. 2017-10-02. On the smallest snarks with oddness 4 and connectivity 2. https://arxiv.org/abs/1710.00757

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