Coloring planar Toeplitz graphs and the stable set polytope

Abstract

Cliques and odd cycles are well known to induce facet-defining 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 first part of this paper we generalize both notions to ( K n ⧹ e)-cycles, a new class of n-critical graphs, and discuss some implications for the class of infinite planar Toeplitz graphs. More precisely, we show that any infinite Toeplitz graph decomposes into a finite number of connected and isomorphic components. Similar to the bipartite case, infinite planar Toeplitz graphs can be characterized by a simple condition on their defining 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 König-type characterization of this class by means of ( K 4⧹ e)-cycles. In the second part, we turn to polyhedral theory and show that ( K n ⧹ e)-cycles give rise to a new class of facet-defining inequalities for the stable set polytope. Then we show how Hajós’ construction can be used to further generalize ( K n ⧹ e)-cycles thereby providing a very large class of n-critical graphs which are still facet-inducing for the associated stable set polytope.