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(v), r} different color classes. The rdynamic chromatic number of a graph G, written χr(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 whenmn ≡ 2 mod 4. This completes the determination of the r-dynamic chromatic number of the m-by-n grid for all r,m, n.