Counting Small Cycle Double Covers

TL;DR

This paper advances the understanding of cycle double covers in planar 4-connected graphs and cubic graphs, establishing linear bounds on the number of small CDCs and providing new proofs and constructions.

Contribution

It proves that such graphs contain linearly many small CDCs, strengthens existing lemmas, and offers a shorter proof for a key CDC bound in cubic graphs.

Findings

Planar 4-connected graphs have linearly many small CDCs.

Planar 4-connected triangulations have stronger CDC results.

Every planar 2-connected cubic graph on n > 4 vertices has a CDC of size at most n/2.

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.