Accessibility setting

Section: New Results

Computing the Chromatic index and clique numberof special graphs

In our paper [17] on the strong chromatic index of planar graphs with large girth, we prove that every planar graph with maximum degree $\Delta$ (let $\Delta$ be an integer) and girth at least $10\Delta +46$ is strong $(2\Delta -1)$-edge-colorable, that is best possible (in terms of number of colors) as soon as $G$ contains two adjacent vertices of degree $\Delta$. This improves the best previous result when $\Delta \ge 6$. In [18] we show how one can compute the clique number of a-perfect graphs in polynomial time. A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovasz and Schrijver 1981). This result relies on polyhedral characterizations of perfect graphs involving the stable set polytope of the graph, a linear relaxation defined by clique constraints, and a semi-definite relaxation, the Theta-body of the graph. A natural question is whether the algorithmic results for perfect graphs can be extended to graph classes with similar polyhedral properties. In [18] we consider a superclass of perfect graphs, the a-perfect graphs, whose stable set polytope is given by constraints associated with generalized cliques. We show that for such graphs the clique number can be computed in polynomial time as well. The result strongly relies upon Fulkersons’s antiblocking theory for polyhedra and Lovasz’s Theta function.