On r-dynamic coloring of grids

An 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(ν), r} different color classes. The r-dynamic chromatic number of a graph G, written χ<inf>r</inf> (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 = 2 mod 4 (for m, n ≥ 3). This completes the determination of the r-dynamic chromatic number of the m-by-n grid for all r, m, n.