Circular Chromatic Number of Signed Graphs

Abstract

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$.  We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular,  we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera.