A note on shortest circuit cover of 3-edge colorable cubic signed graphs

TL;DR

This paper proves that every flow-admissible 3-edge colorable cubic signed graph has a sign-circuit cover whose length is at most approximately 2.22 times the number of edges, advancing understanding of circuit covers in signed graphs.

Contribution

It establishes an upper bound on the length of shortest sign-circuit covers for a specific class of signed graphs, a problem previously studied but not fully resolved.

Findings

Shortest sign-circuit cover length ≤ (20/9) * |E(G)|

Applicable to flow-admissible 3-edge colorable cubic signed graphs

Advances bounds in signed graph circuit cover theory

Abstract

A {sign-circuit cover} $\mathcal{F}$ of a signed graph $(G, \sigma)$ is a family of sign-circuits which covers all edges of $(G, \sigma)$. The shortest sign-circuit cover problem was initiated by M\'a$\check{\text{c}}$ajov\'a, Raspaud, Rollov\'a, and \v{S}koviera (JGT 2016) and received many attentions in recent years. In this paper, we show that every flow-admissible 3-edge colorable cubic signed graph $(G, \sigma)$ has a sign-circuit cover with length at most $\frac{20}{9} |E(G)|$.