On r-dynamic Coloring of Grids

TL;DR

This paper determines the exact $r$-dynamic chromatic number of grid graphs for all parameters, resolving a conjecture and completing the classification for $m$-by-$n$ grids.

Contribution

It proves a conjecture by showing that certain grid graphs cannot have a 3-dynamic 4-coloring when the product of dimensions satisfies a specific modular condition.

Findings

No 3-dynamic 4-coloring exists for $m$-by-$n$ grids when $mn ot\equiv 2 \mod 4$.

Complete characterization of the $r$-dynamic chromatic number for all grid graphs.

Resolved a conjecture and finalized the classification of grid graph colorings.

Abstract

An \textit{$r$-dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of $G$ such that every vertex in $V(G)$ has neighbors in at least $\min\{d(v),r\}$ different color classes. The \textit{$r$-dynamic chromatic number} of a graph $G$, written $\chi_r(G)$, is the least $k$ such that $G$ has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the $m$-by-$n$ grid has no $3$-dynamic $4$-coloring when $mn\equiv2\mod 4$. This completes the determination of the $r$-dynamic chromatic number of the $m$-by-$n$ grid for all $r,m,n$.