(k,1)(k,1)-coloring of sparse graphs

A graph GG is (k,1)(k,1)-colorable   if the vertex set of GG can be partitioned into subsets V1V1 and V2V2 such that the graph G[V1]G[V1] induced by the vertices of V1V1 has maximum degree at most kk and the graph G[V2]G[V2] induced by the vertices of V2V2 has maximum degree at most 11. We prove that every graph with maximum average degree less than 10k+223k+9 admits a (k,1)(k,1)-coloring, where k≥2k≥2. In particular, every planar graph with girth at least 7 is (2,1)(2,1)-colorable, while every planar graph with girth at least 6 is (5,1)(5,1)-colorable. On the other hand, when k≥2k≥2 we construct non-(k,1)(k,1)-colorable graphs whose maximum average degree is arbitrarily close to 14k4k+1.