On the smallest snarks with oddness 4 and connectivity 2

TL;DR

This paper corrects a previous claim by identifying all smallest snarks with oddness 4 up to 34 vertices, including a previously missing example on 28 vertices, using exhaustive computer search.

Contribution

It provides the first complete enumeration of all snarks with oddness 4 up to 34 vertices, correcting earlier inaccuracies and adding a missing example.

Findings

Identified three snarks with oddness 4 on 28 vertices.

Enumerated all snarks with oddness 4 up to 34 vertices.

Corrected previous claims about the smallest such snarks.

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.