On r-dynamic coloring of graphs

An r-dynamic proper 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 r-dynamic chromatic number of a graph G, written χr(G), is the least k such that G has such a coloring. By a greedy coloring algorithm, χr(G) ≤ r∆(G) + 1; we prove that equality holds for ∆(G) > 2 if and only if G is r-regular with diameter 2 and girth 5. We improve the bound to χr(G) ≤ ∆(G) + 2r − 2 when δ(G) > 2r lnn and χr(G) ≤ ∆(G) + r when δ(G) > r2 lnn. In terms of the chromatic number, we prove χr(G) ≤ rχ(G) when G is k-regular with k ≥ (3 + o(1))r ln r and show that χr(G) may exceed r1.377χ(G) when k = r. We prove χ2(G) ≤ χ(G) + 2 when G has diameter 2, with equality only for complete bipartite graphs and the 5-cycle. Also, χ2(G) ≤ 3χ(G) when G has diameter 3, which is sharp. However, χ2 is unbounded on bipartite graphs with diameter 4, and χ3 is unbounded on bipartite graphs with diameter 3 or 3-colorable graphs with diameter 2. Finally, we study χr on grids and toroidal grids.