(k, 1)-coloring of Sparse Graphs

A graph G is called (k, 1)-colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)-coloring, where k ≥ 2. In particular, every planar graph with girth at least 7 is (2, 1)-colorable, while every planar graph with girth at least 6 is (5, 1)-colorable. On the other hand, for each k ≥ 2 we construct non-(k, 1)-colorable graphs whose maximum average degree is arbitrarily close to 14k 4k+1 .