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 in G[V1] every vertex has degree at most j and in G[V2] every vertex 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, φj,k(G), of the difference φj,k(W,G) = ( 2− k+2 (j+2)(k+1) ) |W |−|E(G[W ])| over all subsetsW of V (G). Namely, every graphGwithφj,k(G) > −1 k+1 is (j, k)-colorable. On the other hand, we construct infinitely many non-(j, k)-colorable graphs G with φj,k(G) = −1 k+1 .