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Keywords:

  • colorings;
  • improper colorings;
  • maximum average degree

Abstract

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