Circular Chromatic Number of Signed Graphs

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