The cycle double cover conjecture from the perspective of percolation theory on iterated line graphs

Abstract

The cycle double cover conjecture is a long standing problem in graph theory, which links local properties, the valency of a vertex and no bridges, and a global property of the graph, being covered by a particular set of cycles. We prove the conjecture using a lift of walks and cycles in $G$ to sets of open and closed edges on $\mathcal{L}(\mathcal{L}(G))$, the line graph of the line graph of $G$. We exploit that triangles are preserved by the line graph operator to obtain a one-to-one mapping from walks in the underlying graph $G$ to walks on $\mathcal{L}(\mathcal{L}(G))$. We prove that each set of "double walk covers" in $G$ induces a certain set of $\lbrace 0,1\rbrace$ labels on a subgraph covering of $\mathcal{L}(\mathcal{L}(G))$, minus a set of triangles, and conversely, that there is such a set of labels such that its projection back to $G$ implies a double cycle cover, if $G$ is an simple bridgeless triangle-free cubic graph. The techniques applied are inspired by percolation theory, flipping the $\lbrace 0,1\rbrace$ labels to obtain the desired structure.