Bound on shortest cycle covers
Deping Song, Xuding Zhu
TL;DR
This paper addresses the shortest cycle cover problem in bridgeless graphs by proving a new upper bound $cc(G) < \frac{29}{18}m + \frac{1}{18}n_2$, with the immediate consequence $cc(G) \le \frac{5}{3}m - \frac{1}{42}\log m$. The authors develop a flow-based approach centered on $\mathbb{Z}_2\times\mathbb{Z}_2$-flows and a key lemma that links circuit structure, threads, and zeros of the flow to tighten cycle-cover contributions, leveraging a reduction to cubic bridgeless graphs. They decompose the graph via a large cycle $F$ obtained from a perfect-matchings framework, extend flows so that $E(G)-F$ lies in the flow's support, and construct two cycle covers to bound the total length, yielding the main bound and the log-term corollary. The work also analyzes the impact of the (nowhere-zero) 5-flow conjecture, deriving stronger asymptotic bounds under that hypothesis and highlighting the role of graph structure in improving cycle-cover lengths. Overall, the results push the known limits on cycle-cover lengths and deepen the connection between nowhere-zero flows and efficient edge coverings.
Abstract
Assume $G$ is a bridgeless graph. A cycle cover of $G$ is a collection of cycles of $G$ such that each edge of $G$ is contained in at least one of the cycles. The length of a cycle cover of $G$ is the sum of the lengths of the cycles in the cover. The minimum length of a cycle cover of $G$ is denoted by $cc(G)$. It was proved independently by Alon and Tarsi and by Bermond, Jackson, and Jaeger that $cc(G)\le \frac{5}{3}m$ for every bridgeless graph $G$ with $m$ edges. This remained the best-known upper bound for $cc(G)$ for 40 years. In this paper, we prove that if $G$ is a bridgeless graph with $m$ edges and $n_2$ vertices of degree $2$, then $cc(G) < \frac{29}{18}m+ \frac 1{18}n_2$. As a consequence, we show that $cc(G) \le \frac 53 m - \frac 1{42} \log m$. The upper bound $ cc(G) < \frac{29}{18}m \approx 1.6111 m$ for bridgeless graphs $G$ of minimum degree at least 3 improves the previous known upper bound $1.6258m$. A key lemma used in the proof confirms Fan's conjecture that if $C$ is a circuit of $G$ and $G/C$ admits a nowhere zero 4-flow, then $G$ admits a 4-flow $f$ such that $E(G)-E(C)\subseteq \text{supp} (f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$.