Girth, oddness, and colouring defect of snarks

The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect $0$ if and only if it is $3$-edge-colourable, this invariant can measure how much a cubic graph differs from a $3$-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a $2$-factor. We show that there exist cyclically $5$-edge-connected snarks (cubic graphs with no $3$-edge-colouring) of oddness $2$ and arbitrarily large colouring defect. This result is achieved by means of a construction of cyclically $5$-edge-connected snarks with oddness $2$ and arbitrarily large girth. The fact that our graphs are cyclically $5$-edge-connected significantly strengthens a similar result of Jin and Steffen (2017), which only guarantees graphs with cyclic connectivity at most $3$. At the same time, our result improves Kochol's original construction of snarks with large girth (1996) in that it provides infinitely many nontrivial snarks of any prescribed girth $g\ge 5$, not just girth at least~$g$.