On the chromatic numbers of signed triangular and hexagonal grids (Q2234782)

A signed graph \(G\) consists of a graph \(G = (V,E)\) together with a signature (considered as an edge 2-coloring) \(s: E(G) \to \{1, -1\}\). A homomorphism from a signed graph \(G\) to a signed graph \(H\) is a mapping \(\phi: V(G) \to V(H)\) such that, after performing switching operations at a subset of vertices in \(V(G)\), every edge of \(G\) is mapped to an edge of \(H\) with the same sign. The chromatic number \(\chi_s(G)\) of a signed graph \(G\), is the order of a smallest signed graph \(H\) such that there exists a homomorphism from \(G\) to \(H\). It is shown that the chromatic number of a signed hexagonal grid is 4 and the chromatic number of a signed triangular grid is at most 10. At the end of the paper, an interesting conjectured is posed that the chromatic number of a signed triangular grid equals 6.