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Coloration d'hypergraphes et clique-coloration

Abstract : This work concerns several problems of colorings of hypergraphs, some of which are related to perfect graphs.

We first look at the problem in a global way, and we prove a conjecture due to Sterboul (which generalizes a previous result of Fournier and Las Vergnas) which states that if a hypergraph does not contain a particular type of odd cycle, then it is 2-colorable.

Then we study more precisely the problem of clique-coloring: a maximal clique of a graph is a complete subgraph, maximal by inclusion. The problem consists in assigning colors to the vertices of the graph such that every maximal clique contains at least two vertices with distinct colors. The work of this thesis was originally motivated by the following question : does there exists a constant k such that all perfect graphs are k-clique-colorable? This question is still unsolved, whereas we do not know any odd-hole-free graph that is not 3-clique-colorable. However such a constant exists in many subcases, some of which (such as diamond-free graphs or bull-free graphs) are studied in this thesis.

We also look at the complexity of the problem of clique-coloring, and we try to determine the exact complexity class for every subcases. Several results are proved, especially concerning the complexity of deciding if a perfect graph is 2-clique-colorable: this problem is Sigma_2 P-complete, and is NP-complete for K_4-free perfect graphs.
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Contributor : David Défossez <>
Submitted on : Thursday, November 2, 2006 - 2:51:17 PM
Last modification on : Friday, November 6, 2020 - 4:15:06 AM
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  • HAL Id : tel-00110913, version 1




David Défossez. Coloration d'hypergraphes et clique-coloration. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2006. Français. ⟨tel-00110913⟩



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