The Incidence Chromatic Number of Toroidal Grids

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 $T_{m,n}=C_m\square C_n$ equals 5 when $m,n\equiv 0(\mathrm{mod}5)$ and 6 otherwise.

16 pages