A strong snark is a 2-connected cubic graph which is not 3-edge-colourable and remains so after deleting any edge and suppressing the resulting 2-valent vertices. Strong snarks were introduced by Jaeger in 1985 as a class of cubic graphs that might include counterexamples to the cycle double cover conjecture, the 5-flow conjecture, or to other related longstanding conjectures. With these conjectures still widely open, strong snarks merit further investigation. In this paper we study colouring defect of strong snarks, an invariant introduced by Steffen in 2015 as the minimum number of edges of a cubic graph left uncovered by any set of three perfect matchings. This invariant provides one of measures of edge uncolourability of cubic graphs recently studied by several authors. Our main result shows that the colouring defect of a strong snark is at least 6, and that the bound is sharp.

A longstanding conjecture of Alon and Tarsi, and independly Jaeger (1985), suggests that the edges of every bridgeless graph can be covered with cycles of total length at most 7/5 • m , where m is the number of edges. We study the relationship between cycle covers and structural properties of cubic graphs, focusing on their colouring defect. This invariant, introduced by Steffen in 2015, is defined as the minimum number of edges left uncovered by any set of three perfect matchings of a cubic graph. We show that every bridgeless cubic graph with colouring defect not exceeding 3 admits a cycle cover of length at most 4/3 • m + 1, just one step above the universal lower bound of 4/3 • m for all cubic graphs. We also prove that, regardless of defect, the same bound holds for bridgeless cubic graphs that have an edge whose endvertices removed yield a 3-edge-colourable graph and the edge lies on a 5-cycle. Motivated by our investigations, we introduce a new invariant for cubic graphs, their covering excess, to measure the deviation of the length of a shortest cycle cover from the mentioned lower bound. Finally, we show that every bridgeless cubic graph with covering excess at most 1 admits a cycle double cover.