On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs

Detecting 2-joins faster

2-joins are edge cutsets that naturally appear in the decomposition of several classes of graphs closed under taking induced subgraphs, such as balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free graphs. Their detection is needed in several algorithms, and is the slowest step for some of them. The classical method to detect a 2-join takes $O(n^3m)$ time where $n$ is the number of vertices of the input graph and $m$ the number of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that are needed in all of the known algorithms that use 2-joins), the fastest known method takes time $O(n^4m)$. Here, we give an $O(n^2m)$-time algorithm for both of these problems. A consequence is a speed up of several known algorithms

New characterizations of partition functions using connection matrices

In this thesis we expand upon a line of research pioneered by Freedman, Lovász and Schrijver, and Szegedy, that uses algebraic methods to characterize families of partition functions. We introduce two new types of partition functions: skew partition functions and mixed partition functions. We give two algebraic characterizations of skew partition functions and we show that a mixed partition functions satisfy certain algebraic relationships that are related to the invariant of the symmetric group and to the invariant theory of the Orthosymplectic Supergroup. We furthermore give a characterization of real-valued partition functions for 3-graphs and for virtual link diagrams in terms of positive semidefiniteness of the associated connection matrices

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Asymptotics of characters of symmetric groups related to Stanley character formula

We prove an upper bound for characters of the symmetric groups. Namely, we show that there exists a constant a>0 with a property that for every Young diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr \rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n, c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors needed to write \pi\in S_n as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov's and Biane's character formulas and give a new formula for free cumulants of the transition measure.Comment: Version 4: Change of title, shortened to 20 pages. Version 3: 24 pages, the title and the list of authors were changed. Version 2: 14 pages, the title, abstract and the main result were changed. Version 1: 10 pages (mistake in Lemma 7- which is false

Interactions entre les Cliques et les Stables dans un Graphe

This thesis is concerned with different types of interactions between cliques and stable sets, two very important objects in graph theory, as well as with the connections between these interactions. At first, we study the classical problem of graph coloring, which can be stated in terms of partioning the vertices of the graph into stable sets. We present a coloring result for graphs with no triangle and no induced cycle of even length at least six. Secondly, we study the Erdös-Hajnal property, which asserts that the maximum size of a clique or a stable set is polynomial (instead of logarithmic in random graphs). We prove that the property holds for graphs with no induced path on k vertices and its complement.Then, we study the Clique-Stable Set Separation, which is a less known problem. The question is about the order of magnitude of the number of cuts needed to separate all the cliques from all the stable sets. This notion was introduced by Yannakakis when he studied extended formulations of the stable set polytope in perfect graphs. He proved that a quasi-polynomial number of cuts is always enough, and he asked if a polynomial number of cuts could suffice. Göös has just given a negative answer, but the question is open for restricted classes of graphs, in particular for perfect graphs. We prove that a polynomial number of cuts is enough for random graphs, and in several hereditary classes. To this end, some tools developed in the study of the Erdös-Hajnal property appear to be very helpful. We also establish the equivalence between the Clique-Stable set Separation problem and two other statements: the generalized Alon-Saks-Seymour conjecture and the Stubborn Problem, a Constraint Satisfaction Problem.Cette thèse s'intéresse à différents types d'interactions entre les cliques et les stables, deux objets très importants en théorie des graphes, ainsi qu'aux relations entre ces différentes interactions. En premier lieu, nous nous intéressons au problème classique de coloration de graphes, qui peut s'exprimer comme une partition des sommets du graphe en stables. Nous présentons un résultat de coloration pour les graphes sans triangles ni cycles pairs de longueur au moins 6. Dans un deuxième temps, nous prouvons la propriété d'Erdös-Hajnal, qui affirme que la taille maximale d'une clique ou d'un stable devient polynomiale (contre logarithmique dans les graphes aléatoires) dans le cas des graphes sans chemin induit à k sommets ni son complémentaire, quel que soit k.Enfin, un problème moins connu est la Clique-Stable séparation, où l'on cherche un ensemble de coupes permettant de séparer toute clique de tout stable. Cette notion a été introduite par Yannakakis lors de l’étude des formulations étendues du polytope des stables dans un graphe parfait. Il prouve qu’il existe toujours un séparateur Clique-Stable de taille quasi-polynomiale, et se demande si l'on peut se limiter à une taille polynomiale. Göös a récemment fourni une réponse négative, mais la question se pose encore pour des classes de graphes restreintes, en particulier pour les graphes parfaits. Nous prouvons une borne polynomiale pour la Clique-Stable séparation dans les graphes aléatoires et dans plusieurs classes héréditaires, en utilisant notamment des outils communs à l'étude de la conjecture d'Erdös-Hajnal. Nous décrivons également une équivalence entre la Clique-Stable séparation et deux autres problèmes  : la conjecture d'Alon-Saks-Seymour généralisée et le Problème Têtu, un problème de Satisfaction de Contraintes