# The smallest nontrivial snarks of oddness 4

## TL;DR

This paper classifies all minimal cubic graphs called snarks with oddness 4, cyclic connectivity 4, and 44 vertices, revealing 31 unique structures and their infinite families, combining theoretical proofs with computational verification.

## Contribution

It provides a complete classification of the smallest nontrivial snarks with oddness 4 and cyclic connectivity 4, including their structural properties and infinite families.

## Key findings

- Exactly 31 such snarks exist on 44 vertices.
- All these snarks have girth 5 and are constructed from Petersen subgraphs.
- The 31 snarks form a complete set with proven structural properties.

## Abstract

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs admitting no proper 3-edge-colouring). In [Ars Math. Contemp. 16 (2019), 277-298] we have proved that the smallest number of vertices of a snark with cyclic connectivity 4 and oddness 4 is 44. We now show that there are exactly 31 such snarks, all of them having girth 5. These snarks are built up from subgraphs of the Petersen graph and a small number of additional vertices. Depending on their structure they fall into six classes, each class giving rise to an infinite family of snarks with oddness at least 4 with increasing order. We explain the reasons why these snarks have oddness 4 and prove that the 31 snarks form the complete set of snarks with cyclic connectivity 4 and oddness 4 on 44 vertices. The proof is a combination of a purely theoretical approach with extensive computations performed by a computer.