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.

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A signed graph is a pair (G, σ), where G is a graph and σ : E(G) → {+, −} 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, σ) a circular r-coloring of (G, σ) is an assignment ψ of points of a circle of circumference r to the vertices of G such that for every edge e = uv of G, if σ(e) = +, then ψ(u) and ψ(v) have distance at least 1, and if σ(e) = −, then ψ(v) and the antipodal of ψ(u) have distance at least 1. The circular chromatic number χ c (G, σ) of a signed graph (G, σ) is the infimum of those r for which (G, σ) admits a circular r-coloring. For a graph G, we define the signed circular chromatic number of G to be max{χ c (G, σ) : σ is a signature of G}.We study basic properties of circular coloring of signed graphs and develop tools for calculating χ c (G, σ). 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 + 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.

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A signed bipartite (simple) graph (G, σ) is said to be C −4 -critical if it admits no homomorphism to C −4 (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C −4 . In particular, the 4-color theorem is equivalent to: Given a planar graph G, the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to C −4 .We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any C −4 -critical signed graph on n vertices must have at least ⌈ 4n 3 ⌉ edges, and that this bound or ⌈ 4n 3 ⌉ + 1 is attained for each value of n ≥ 9. As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to C −4 . Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to C −4 , showing 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the above mentioned restatement of the 4CT.