We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! ? 1-cliques in minimal imperfect graphs.

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The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Claude Berge made a conjecture about them, that was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and is now called the strong perfect graph theorem. This is a survey about perfect graphs, mostly focused on the strong perfect gr...

This paper first extends the result of Blakley and Kabatianski [3] to general non-perfect SSS using information-theoretic arguments. Furthermore, we refine Okada and Kurosawa’s lower bound [12] into a much more precise information-theoretic characterization of non-perfect secret sharing idealness. In the light of this generalization, we establish that ideal schemes do not always have a matroida...

Grouping datasets plays an important role in many scientific researches. Depending on data features and applications, different constrains are imposed on groups, while having groups with similar members is always a main criterion. In this paper, we propose an algorithm for grouping the objects with random labels, nominal features having too many nominal attributes. In addition, the size constra...