Counting Circuit Double Covers

Several recent results and conjectures study counting versions of classical existence statements. We ask the same question for circuit double covers of cubic graphs. We prove an exponential bound for planar graphs: Every bridgeless cubic planar graph with n vertices has at least \((5/2)^{n/4 - 1/2}\) circuit double covers. The method we used to obtain this bound motivates a general framework for counting objects on graphs using linear algebra which might be of independent interest. We also conjecture that every bridgeless cubic graph has at least \(2^{n/2-1}\) circuit double covers.