Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

Abstract

A circular r-coloring of a signed graph (G,σ) is to assign points of a circle of circumference r, r≥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 r2−1. The restriction of the notion of circular colorings to the class of signed bipartite graphs is already of high interest because the circular chromatic number of an (unsigned) graph can be obtained by bounding the circular chromatic number of an associated signed bipartite graph.In this paper, we define signed bipartite circular cliques Bp;qs and Bˆp;qs having the property that a signed bipartite graph admits a circular pq-coloring if and only if it admits an edge-sign preserving homomorphism to Bp;qs and a switching homomorphism to Bˆp;qs, respectively. Then as a bipartite analogue of Grötzsch's theorem, we prove that every signed bipartite planar graph of negative girth at least 6 admits a circular 3-coloring.