Signed Circuit $6$-Covers of Signed $K_4$-Minor-Free Graphs

Abstract

Bermond, Jackson and Jaeger [J. Combin. Theory Ser. B 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [J. Combin. Theory Ser. B 54 (1992): 113-122] showed that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit $k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is the well-know circuit double cover conjecture. For signed circuit $k$-cover of signed graphs, it is known that for every integer $k\leq 5$, there are infinitely many coverable signed graphs without signed circuit $k$-cover and there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't admit a signed circuit $1$-cover. Fan conjectured that every coverable signed graph has a signed circuit $6$-cover. This conjecture was verified only for signed eulerian graphs and for signed graphs whose bridgeless-blocks are eulerian. In this paper, we prove that this conjecture holds for signed $K_4$-minor-free graphs. The $6$-cover is best possible for signed $K_4$-minor-free graphs.