2-Edge-Colored Chromatic Number of Grids is at Most 9

Abstract

AbstractA 2-edge-colored graph is a pair $$(G, \sigma )$$(G,σ) where G is a graph, and $$\sigma :E(G)\rightarrow \{\text {'}+\text {'},\text {'}-\text {'}\}$$σ:E(G)→{'+','-'} is a function which marks all edges with signs. A 2-edge-colored coloring of the 2-edge-colored graph $$(G, \sigma )$$(G,σ) is a homomorphism into a 2-edge-colored graph $$(H, \delta )$$(H,δ). The 2-edge-colored chromatic number of the 2-edge-colored graph $$(G, \sigma )$$(G,σ) is the minimum order of H. In this paper we show that for every 2-dimensional grid $$(G, \sigma )$$(G,σ) there exists a homomorphism from $$(G, \sigma )$$(G,σ) into the 2-edge-colored Paley graph $$SP_9$$SP9. Hence, the 2-edge-colored chromatic number of the 2-edge-colored grids is at most 9. This improves the upper bound on this number obtained recently by Bensmail. Additionally, we show that 2-edge-colored chromatic number of the 2-edge-colored grids with 3 columns is at most 8.