Signified chromatic number of grids is at most 9
A signified graph is a pair $(G, Σ)$ where $G$ is a graph, and $Σ$ is a set of edges marked with '$-$'. Other edges are marked with '$+$'. A signified coloring of the signified graph $(G, Σ)$ is a homomorphism into a signified graph $(H, Δ)$. The signified chromatic number of the signified graph $(G, Σ)$ is the minimum order of $H$. In this paper we show that for every 2-dimensional grid $(G, Σ)$ there exists homomorphism from $(G, Σ)$ 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.
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