On the smallest snarks with oddness 4 and connectivity 2
Combinatorics
2018-04-30 v2 Discrete Mathematics
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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Cite
@article{arxiv.1710.00757,
title = {On the smallest snarks with oddness 4 and connectivity 2},
author = {Jan Goedgebeur},
journal= {arXiv preprint arXiv:1710.00757},
year = {2018}
}
Comments
5 pages