Limits of Near-Coloring of Sparse Graphs

Let a, b, d be non-negative integers. A graph G is (d, a, b)-colorable if its vertex set can be partitioned into a + b sets D1, . . . , Da, O1, . . . , Ob such that the graph G[Di] induced by Di has maximum degree at most d for 1 ≤ i ≤ a, while the graph G[Oj ] induced by Oj is an edgeless graph for 1 ≤ j ≤ b. In this paper, we give two real-valued functions f and g such that any graph with maximum average degree at most f(d, a, b) is (d, a, b)-colorable, and there exist non-(d, a, b)-colorable graphs with average degree at most g(d, a, b). Both these functions converge (from below) to 2a+ b when d tends to infinity. This implies that allowing a color to be d-improper (i.e. of typeDi) even for a large degree d increases the maximum average degree that guarantees the existence of a valid coloring only by 1. Using a color of typeDi (even with a very large degree d) is somehow less powerful than using two colors of type Oj (two stable sets).