Signed bipartite circular cliques and a bipartite analogue of Gr\"{o}tzsch's theorem

Abstract

The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, \sigma)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, \sigma)$ admits a circular $r$-coloring is said to be the circular chromatic number of $(G, \sigma)$ and is denoted by $\chi_c(G, \sigma)$. For any rational number $r=\frac{p}{q}$, two notions of circular cliques are presented corresponding to the edge-sign preserving homomorphism and the switching homomorphism. It is also shown that the restriction of the study of circular chromatic numbers to the class of signed bipartite simple graphs already captures the study of circular chromatic numbers of graphs via basic graph operations, even though the circular chromatic number of every signed bipartite graph is bounded above by $4$. In this work, we consider the restriction of the circular chromatic number to this class of signed graphs and construct signed bipartite circular cliques with respect to both notions of homomorphisms. We then present reformulations of the $4$-Color Theorem and the Gr\"otzsch theorem. As a bipartite analogue of Gr\"otzsch's theorem, we prove that every signed bipartite planar graph of negative girth at least $6$ has circular chromatic number at most $3$.