Contract grant sponsor: French Agence Nationale de la Recherche; Contract grant numbers: anr-09-jcjc-0041-01; anr-10-jcjc-0204-01; Contract grant sponsor: LaBRI (P. O.).
A Complexity Dichotomy for the Coloring of Sparse Graphs
Article first published online: 26 JUN 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 73, Issue 1, pages 85–102, May 2013
How to Cite
Esperet, L., Montassier, M., Ochem, P. and Pinlou, A. (2013), A Complexity Dichotomy for the Coloring of Sparse Graphs. J. Graph Theory, 73: 85–102. doi: 10.1002/jgt.21659
- Issue published online: 5 MAR 2013
- Article first published online: 26 JUN 2012
- Manuscript Revised: 31 JAN 2012
- Manuscript Received: 5 JAN 2011
- French Agence Nationale de la Recherche. Grant Number: anr-09-jcjc-0041-01; anr-10-jcjc-0204-01
- LaBRI (P. O.)
- sparse graphs
Galluccio, Goddyn, and Hell proved in 2001 that in any minor-closed class of graphs, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. Let be a monotone class of graphs containing all planar graphs, and closed under clique-sum of order at most two. Examples of such class include minor-closed classes containing all planar graphs, and such that all minimal obstructions are 3-connected. We prove that for any k and g, either every graph of girth at least g in has a homomorphism to , or deciding whether a graph of girth g in has a homomorphism to is NP-complete. We also show that the same dichotomy occurs when considering 3-Colorability or acyclic 3-Colorability of graphs under various notions of density that are related to a question of Havel (On a conjecture of Grünbaum, J Combin Theory Ser B 7 (1969), 184–186) and a conjecture of Steinberg (The state of the three color problem, Quo Vadis, Graph theory?, Ann Discrete Math 55 (1993), 211–248) about the 3-Colorability of sparse planar graphs.