On the chromatic numbers of signed triangular and hexagonal grids

TL;DR

This paper investigates the chromatic numbers of signed triangular and hexagonal grids, establishing upper bounds of 10 and 4 respectively, through homomorphism-based analysis of signed graphs.

Contribution

It provides new upper bounds for the chromatic numbers of signed triangular and hexagonal grids, advancing understanding of coloring properties in signed graph classes.

Findings

Chromatic number of signed triangular grids is at most 10.

Chromatic number of signed hexagonal grids is at most 4.

Homomorphism approach used to derive bounds.

Abstract

A signed graph is a simple graph with two types of edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $\varphi: V(G) \rightarrow V(H)$ such that, after switching any number of the vertices of $G$, $\varphi$ maps every edge of $G$ to an edge of the same type in $H$. The chromatic number $\chi_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. We show that the chromatic number of signed triangular grids is at most 10 and the chromatic number of signed hexagonal grids is at most 4.