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arXiv · 2506.10604

Counting Small Cycle Double Covers

Jorik Jooken·Ben Seamone·Carol T. Zamfirescu
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Abstract

A theorem due to Seyffarth states that every planar $4$-connected $n$-vertex graph has a cycle double cover (CDC) containing at most $n-1$ cycles (a "small" CDC). We extend this theorem by proving that, in fact, such a graph must contain linearly many small CDCs (in terms of $n$), and provide stronger results in the case of planar $4$-connected triangulations. We complement this result with constructions of planar $4$-connected graphs which contain at most polynomially many small CDCs. Thereafter we treat cubic graphs, strengthening a lemma of Hu\v{s}ek and \v{S}\'amal on the enumeration of CDCs, and, motivated by a conjecture of Bondy, give an alternative proof of the result that every planar 2-connected cubic graph on $n > 4$ vertices has a CDC of size at most $n/2$. Our proof is much shorter and obtained by combining a decomposition based argument, which might be of independent interest, with further combinatorial insights. Some of our results are accompanied by a version thereof for CDCs containing no cycle twice.

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Jorik Jooken, Ben Seamone, Carol T. Zamfirescu. 2025-06-12. Counting Small Cycle Double Covers. https://arxiv.org/abs/2506.10604

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