Shortest circuit covers of signed graphs

Abstract

A shortest circuit cover $\mathcal{F}$ of a bridgeless graph $G$ is a family of circuits that covers every edge of $G$ and is of minimum total length. The total length of a shortest circuit cover $\mathcal{F}$ of $G$ is denoted by $SCC(G)$. For ordinary graphs (graphs without sign), the subject of shortest circuit cover is closely related to some mainstream areas, such as, Tutte's integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and others. For signed graphs $G$, it is proved recently by Máčajová, Raspaud, Rollová and Škoviera that $SCC(G)\le 11|E|$ if $G$ is s-bridgeless, and $SCC(G)\le 9|E|$ if $G$ is $2$-edge-connected. In this paper this result is improved as follows, $$SCC(G)\le |E|+3|V|+z$$ where $z=\min \{\frac{2}{3}|E|+\frac{4}{3}{\epsilon }_N-7,|V|+2{\epsilon }_N-8\}$ and ${\epsilon }_N$ is the negativeness of $G$. The above upper bound can be further reduced if $G$ is $2$-edge-connected with even negativeness.

Cited by in corpus (2)

• Nowhere-zero flows in signed graphs: A survey
• Minimum $T$-Joins and Signed-Circuit Covering