Signified chromatic number of grids is at most 9

09/01/2019

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by   Janusz Dybizbański, et al.

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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.