A Complexity Dichotomy for the Coloring of Sparse Graphs

Gallucio, Goddyn and Hell proved in 2001 that in any minor-closed class, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. We show that for any minor-closed class F containing all planar graphs, and such that all minimal obstructions are 3-connected, the following holds: for any k there is a g = g(F , k) such that every graph of girth at least g in F has a homomorphism to C2k+1, but deciding whether a graph of girth g − 1 in F has a homomorphism to C2k+1 is NP-complete. The classes of graphs on which this result applies include planar graphs, Kn-minor free graphs, and graphs with bounded Colin de Verdière parameter (for instance, linklessly embeddable graphs). We also show that the same dichotomy occurs in problems related to a question of Havel (1969) and a conjecture of Steinberg (1976) about the 3-colorability of sparse planar graphs. This work was partially supported by the French Agence Nationale de la Recherche, through projects GRATOS (anr-09-jcjc-0041-01) and HEREDIA (anr-10-jcjc-020401). Laboratoire G-SCOP (Grenoble-INP, CNRS), Grenoble, France. LaBRI (Université de Bordeaux, CNRS), Talence, France. LRI (Université Paris Sud, CNRS), Orsay, France. LIRMM (Université Montpellier 2, CNRS), Montpellier, France.