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