On [formula omitted]-dynamic coloring of grids

Abstract

An r-dynamick-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 r-dynamic 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 when mn≡2mod4 (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.