The Incidence Chromatic Number of Toroidal Grids

TL;DR

This paper determines the exact incidence chromatic number of toroidal grids, showing it is 5 when both dimensions are multiples of 5 and 6 otherwise, advancing understanding of graph coloring in grid graphs.

Contribution

The paper provides a complete characterization of the incidence chromatic number for toroidal grids based on their dimensions.

Findings

Incidence chromatic number is 5 when both dimensions are divisible by 5.

Incidence chromatic number is 6 in all other cases.

Exact values depend on divisibility conditions of grid dimensions.

Abstract

An incidence in a graph $G$ is a pair $(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$ are incident. Two incidences $(v,e)$ and $(w,f)$ are adjacent if $v=w$, or $e=f$, or the edge $vw$ equals $e$ or $f$. The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\Box C_n$ equals 5 when $m,n \equiv 0 \pmod 5$ and 6 otherwise.