Cubic graphs of colouring defect 3 and conjectures of Berge and Alon-Tarsi

Abstract: We study two measures of uncolourability of cubic graphs, their colouring defect and perfect matching index. The colouring defect of a cubic graph $G$ is the smallest number of edges left uncovered by three perfect matchings; the perfect matching index of $G$ is the smallest number of perfect matchings that together cover all edges of $G$. We provide a complete characterisation of cubic graphs with colouring defect $3$ whose perfect matching index is greater or equal to $5$. The result states that every such graph arises from the Petersen graph with a fixed $6$-cycle $C$ by substituting edges or vertices outside $C$ with suitable $3$-edge-colourable cubic graphs. Our research is motivated by two deep and long-standing conjectures, Berge's conjecture stating that five perfect matchings are enough to cover the edges of any bridgeless cubic graph and the shortest cycle cover conjecture of Alon and Tarsi suggesting that every bridgeless graph can have its edges covered with cycles of total length at most $7/5\cdot m$, where $m$ is the number of edges. We apply our characterisation to showing that every cubic graph with colouring defect $3$ admits a cycle cover of length at most $4/3\cdot m +1$, where $m$ is the number of edges, the bound being achieved by the graphs whose perfect matching index equals $5$. We further prove that every snark containing a $5$-cycle with an edge whose endvertices removed yield a $3$-edge-colourable graph has a cycle cover of length at most $4/3\cdot m+1$, as well.