Signified chromatic number of grids is at most 9

Abstract

A signified graph is a pair $(G, \Sigma)$ where $G$ is a graph, and $\Sigma$ is a set of edges marked with '$-$'. Other edges are marked with '$+$'. A signified coloring of the signified graph $(G, \Sigma)$ is a homomorphism into a signified graph $(H, \Delta)$. The signified chromatic number of the signified graph $(G, \Sigma)$ is the minimum order of $H$. In this paper we show that for every 2-dimensional grid $(G, \Sigma)$ there exists homomorphism from $(G, \Sigma)$ into the signed Paley graphs $SP_9$. Hence signified chromatic number of the signified grids is at most 9. This improves upper bound on this number obtained recently by Bensmail.