Brooks' theorem for signed graphs with $\Delta=3$

Abstract

Circular $r$-coloring of a signed graph $(G,\sigma)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a positive connection, the distance of one from the antipodal of the other is at least $1$. A signed graph $(G,\sigma)$ admits a circular $r$-coloring for some values of $r$ if and only if it has no negative loop. The smallest value of such $r$ is the circular chromatic number, denoted $\chi_{c}(G,\sigma)$. The circular chromatic number is a refinement of the balanced chromatic number, which is mostly studied under the equivalent term $0$-free coloring in the literature. Extending Brooks' theorem, M\'a\v cajov\'a, Raspaud, and \v{S}koviera showed that if $\Delta(G)$ is an even number, $G$ is connected, and $(G,\sigma)$ is not (switching) isomorphic to $(K_{\Delta+1},-)$ or $C_{-\ell}$ (when $\Delta(G)=2$), then $\chi_c(G,\sigma)\leq \Delta(G)$ and that the upper bound is tight. For the odd values of $\Delta(G)$, assuming a connected signed graph $(G,\sigma)$ is not isomorphic to $(K_{\Delta+1},-)$, determining the best upper bound for $\chi_c(G, \sigma)$ proves to be more of a challenge. In this work, addressing the first step of this question, we show that if $(G, \sigma)$ is a signed graph of maximum degree 3 with no component isomorphic to $(K_4, -)$, then $\chi_{c}(G, \sigma)\leq \frac{10}{3}$. The upper bound is tight even among signed cubic graphs of girth 5. In particular, there is a signature on the Petersen graph for which the upper of $\frac{10}{3}$ is achieved.