Claw-free circular-perfect graphs

Arnaud Pêcher 1, 2 Xuding Zhu 3
1 Realopt - Reformulations based algorithms for Combinatorial Optimization
LaBRI - Laboratoire Bordelais de Recherche en Informatique, IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : The circular chromatic number of a graph is a well-studied refinement of the chromatic number. Circular-perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This paper studies claw-free circular-perfect graphs. First we prove that if $G$ is a connected claw-free circular-perfect graph with $\chi(G) > \omega(G)$, then $\min\{\alpha(G), \omega(G)\} =2$. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw-free circular-perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs $G$ have $\min \{\alpha(G), \omega(G)\} = 2$. In contrast to this result, it is shown in \cite{PanZhu2006} that minimal circular-imperfect graphs $G$ can have arbitrarily large independence number and arbitrarily large clique number. In this paper, we prove that claw-free minimal circular-imperfect graphs $G$ have $\min \{\alpha(G), \omega(G)\} \leq 3$.
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https://hal.archives-ouvertes.fr/hal-00431241
Contributor : Arnaud Pêcher <>
Submitted on : Wednesday, November 11, 2009 - 8:45:00 AM
Last modification on : Friday, July 12, 2019 - 8:10:02 PM

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Arnaud Pêcher, Xuding Zhu. Claw-free circular-perfect graphs. Journal of Graph Theory, Wiley, 2010, 65 (2), pp.163-172. ⟨10.1002/jgt.20474⟩. ⟨hal-00431241⟩

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