Limits of Near-Coloring of Sparse Graphs


  • Contract grant sponsor: Czech Science Foundation; Contract grant number: P202/12/G061 (T. K.); Contract grant sponsor: ANR; Contract grant number: GRATOS - ANR-09-JCJC-0041-01 (M. M.); Contract grant sponsor: ANR-NSC; Contract grant numbers: GRATEL - ANR-09-blan-0373-01; NSC99-2923-M-110-001-MY3 (A. R.).


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