3 research outputs found
The smallest nontrivial snarks of oddness 4
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.Comment: 38 pages; submitted for publicatio