main.abs

Snarks, that is $2$-connected cubic graphs admitting no $3$-edge-colouring, provide a promising family of cubic graphs with respect to many widely-open conjectures. They are often constructed by joining several building blocks which can be regarded as cubic ``graphs'' with dangling edges allowed, formally called multipoles. Colouring properties of multipoles with at most five dangling edges relevant for constructions of snarks are almost completely characterised. The remaining missing class of such multipoles are so-called proper (2,3)-poles that can be obtained by severing an edge and removing a vertex from a snark.

    Therefore, in our work, we analyse the colouring properties of proper (2,3)-poles. To conduct our analysis, we explore all proper (2,3)-poles resulting from nontrivial snarks with at most 28 vertices. This encompasses a~total of 3,247 snarks and 3,476,400 proper (2,3)-poles. In our research, we provide various structures that can be utilized to expand the colourability of proper (2,3)-poles. In the core of our work, we provide theorems regarding the colouring properties of proper (2,3)-poles, specifically necessary and sufficient conditions for these properties. Additionally, we present the data and observations from the analysis.