Coloring planar Toeplitz graphs and the stable set polytope

Cliques and odd cycles are well known to induce facet-de.ning inequalities for the stable set polytope. In graph coloring cliques are a class of n-critical graphs whereas odd cycles represent the class of 3-critical graphs. In the .rst part of this paper we generalize both notions to (Kn \ e)-cycles, a new class of n-critical graphs, and discuss some implications for the class of in.nite planar Toeplitz graphs. More precisely, we show that any in.nite Toeplitz graph decomposes into a .nite number of connected and isomorphic components. Similar to the bipartite case, in.nite planar Toeplitz graphs can be characterized by a simple condition on their de.ning 0–1 sequence. We then address the problem of coloring such graphs. Whereas they can always be 4-colored by a greedy-like algorithm, we are able to fully characterize the subclass of 3-chromatic such graphs. As a corollary, we obtain a K5 onig-type characterization of this class by means of (K4 \ e)-cycles. In the second part, we turn to polyhedral theory and show that (Kn \ e)-cycles give rise to a new class of facet-de.ning inequalities for the stable set polytope. Then we show how Haj8 os’ construction can be used to further generalize (Kn \ e)-cycles thereby providing a very large class of n-critical graphs which are still facet-inducing for the associated stable set polytope. c © 2003 Elsevier B.V. All rights reserved.