Signified chromatic number of grids is at most 9

TL;DR

This paper proves that the signified chromatic number of any 2D grid is at most 9 by demonstrating a homomorphism into the signed Paley graph $SP_9$, improving previous bounds.

Contribution

The paper establishes an upper bound of 9 for the signified chromatic number of 2D grids using homomorphisms into $SP_9$, advancing prior results.

Findings

Signified chromatic number of 2D grids is at most 9.

Homomorphism into signed Paley graph $SP_9$ exists for all 2D grids.

Improves previous upper bounds by Bensmail.

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.