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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 aglobal property of the graph, being covered by a particular set of cycles. Weprove the conjecture using a lift of walks and cycles in $G$ to sets of openand closed edges on $\mathcal{L}(\mathcal{L}(G))$, the line graph of the linegraph of $G$. We exploit that triangles are preserved by the line graphoperator 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 "doublewalk covers" in $G$ induces a certain set of $\lbrace 0,1\rbrace$ labels on asubgraph 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 projectionback to $G$ implies a double cycle cover, if $G$ is an simple bridgelesstriangle-free cubic graph. The techniques applied are inspired by percolationtheory, flipping the $\lbrace 0,1\rbrace$ labels to obtain the desiredstructure.

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