On 11-improper 22-coloring of sparse graphs

A graph G is (1, 1)-colorable if its vertices can be partitioned into subsets V1 and V2 so that every vertex in G[Vi] has degree at most 1 for each i ∈ {1, 2}. We prove that every graph with maximum average degree at most 14 5 is (1, 1)-colorable. In particular, it follows that every planar graph with girth at least 7 is (1, 1)-colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to 14 5 (from above) that are not (1, 1)-colorable. In fact, we establish the best possible sufficient condition for the (1, 1)-colorability of a graph G in terms of the minimum, ρG, of ρG(S) = 7|S|−5|E(G[S])| over all subsets S of V (G). Namely, every graph G with ρG ≥ 0 is (1, 1)-colorable. On the other hand, we construct infinitely many non-(1, 1)-colorable graphs G with ρG = −1.