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.).
Limits of Near-Coloring of Sparse Graphs
Article first published online: 6 FEB 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 75, Issue 2, pages 191–202, February 2014
How to Cite
Dorbec, P., Kaiser, T., Montassier, M. and Raspaud, A. (2014), Limits of Near-Coloring of Sparse Graphs. J. Graph Theory, 75: 191–202. doi: 10.1002/jgt.21731
- Issue published online: 2 DEC 2013
- Article first published online: 6 FEB 2013
- Manuscript Revised: 18 DEC 2012
- Manuscript Received: 28 MAR 2012
- Czech Science Foundation. Grant Number: P202/12/G061
- ANR. Grant Number: ANR-09-JCJC-0041-01
- ANR-NSC. Grant Numbers: ANR-09-blan-0373-01, NSC99-2923-M-110-001-MY3
- improper colorings;
- maximum average degree
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).