# On the Smallest Snarks with Oddness 4 and Connectivity 2

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

### 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, 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.