Circuit Covers of Cubic Signed Graphs

Abstract

A signed graph is a graph $G$ associated with a mapping $σ: E(G)\to \{-1,+1\}$, denoted by $(G,σ)$. A $cycle$ of $(G,σ)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges, and negative otherwise. A $circuit$ of of a signed graph $(G,σ)$ is a positive cycle or a barbell consisting of two edge-disjoint negative cycles joined by a path. The definition of a circuit of signed graph comes from the signed-graphic matroid. A circuit cover of $(G,σ)$ is a family of circuits covering all edges of $(G,σ)$. A circuit cover with the smallest total length is called a shortest circuit cover of $(G,σ)$ and its length is denoted by $\text{scc}(G,σ)$. Bouchet proved that a signed graph with a circuit cover if and only if it is flow-admissible (i.e., has a nowhere-zero integer flow). Máčajová et. al. show that a 2-edge-connected signed graph $(G,σ)$ has $\text{scc}(G,σ)\le 9 |E(G)|$ if it is flow-admissible. This bound was improved recently by Cheng et. al. to $\text{scc}(G,σ) \le 11|E(G)|/3$ for 2-edge-connected signed graphs with even negativeness, and particularly, $\text{scc}(G,σ)\le 3|E(G)|+ε(G,σ)/3$ for 2-edge-connected cubic signed graphs with even negativeness (where $ε(G,σ)$ is the negativeness of $(G,σ)$). In this paper, we show that every 2-edge-connected cubic signed graph has $\text{scc}(G,σ)\le 26|E(G)|/9$ if it is flow-admissible, and $\text{scc}(G,σ)\le 23|E(G)|/9$ if it has even negativeness.

13 pages, 2 figures