On rr-dynamic coloring of grids

An rr-dynamic  kk-coloring   of a graph GG is a proper kk-coloring of GG such that every vertex in V(G)V(G) has neighbors in at least min{d(v),r}min{d(v),r} different color classes. The rr-dynamic chromatic number   of a graph GG, written χr(G)χr(G), is the least kk such that GG has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the mm-by-nn grid has no 3-dynamic 4-coloring when mn≡2mod4mn≡2mod4 (for m,n≥3m,n≥3). This completes the determination of the rr-dynamic chromatic number of the mm-by-nn grid for all r,m,nr,m,n.