Limits of Near-Coloring of Sparse Graphs

Abstract

Let be nonnegative integers. A graph G is -colorable if its vertex set can be partitioned into sets such that the graph induced by has maximum degree at most d for , while the graph induced by is an edgeless graph for . In this article, we give two real-valued functions and such that any graph with maximum average degree at most is -colorable, and there exist non--colorable graphs with average degree at most . Both these functions converge (from below) to when d tends to infinity. This implies that allowing a color to be d-improper (i.e., of type ) 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 type (even with a very large degree d) is somehow less powerful than using two colors of type (two stable sets).