Defective 2-colorings of sparse graphs

A graph G is (j, k)-colorable if its vertices can be partitioned into subsets V₁ and V₂ such that every vertex in G[V₁] has degree at most j and every vertex in G[V₂] has degree at most k. We prove that if k≥2j+2, then every graph with maximum average degree at most 2(2-k+2(j+2)(k+1)) is (j, k)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close to 2(2-k+2(j+2)(k+1)) (from above) that are not (j, k)-colorable. In fact, we prove a stronger result by establishing the best possible sufficient condition for the (j, k)-colorability of a graph G in terms of the minimum, φ_j,k(G), of the difference φj,k(W,G)=(2-k+2(j+2)(k+1))|W|-|E(G[W])| over all subsets W of V(G). Namely, every graph G with φj,k(G)>-1k+1 is (j, k)-colorable. On the other hand, we construct infinitely many non-(j, k)-colorable graphs G with φj,k(G)=-1k+1.