Claude Berge

Packing Problems and Hypergraph Theory: A Survey

Publisher Summary Graph theory has appeared as a tool to solve a large class of combinatorial problems. Hypergraph theory started to generalize and to simplify the classical theorems about graphs. This chapter focuses on a particular type of problems that arise in graph theory and in integer programming; though some of the results can be proved by purely algebraic methods and polytope properties, the “hypergraph” context permits a significant presentation that does not require a particular algebraic knowledge. Most of the significant concepts about graphs are in fact (0, 1) solutions of a linear program; if instead one consider the solutions with fractional coordinates, one can get simpler results. This is called the “fractional” graph theory. The chapter provides a brief overview of classical hypergraph theory and fractional hypergraph theory.

A combinatorial problem in logic

Abstract This short note is an application of some theorems of graph theory to the problem of the minimum number of counter-examples needed to show that a special class of theories is complete.

Coloring the Edges of A Hypergraph and Linear Programming Techniques

A theorem of Baranyai reduces the problem of finding the chromatic index of certain hypergraphs to a cutting stock integer programming problem. Baranyai used this result to establish the chromatic index for the complete h -uniform hypergraphs. We use a linear programming technique of Gomory and Gilmore to extend his result to two other cases: the hereditary closure of the complete h -uniform hypergraphs K h n , for h ≤ 4; and of the complete h -partite hypergraphs.

Minimax relations for the partial q -colorings of a graph

Abstract A partial q-coloring of a graph is a family of q disjoint stable sets, each one representing a “color”; the largest number of colored vertices in a partial q -coloring is a number α q ( G ), extension of the stability number α ( G )= α 1 ( G ). In this note, we investigate the possibilities, for 1⩽ q ⩽ γ ( G ) , to express α q ( G ) by a minimax equality.

On the existence of subgraphs with degree constraints

Let G be a graph on X, and let f(x), g(x) be positive integers; several authors have given conditions for the existence of a graph H on X, obtained from G by removing or duplicating edges, with degrees dH(x) in the intervals [g(x), f(x)] (for short, we shall say that H is a (g, f)-graph). Most of the known conditions are complicated, and our purpose is to show that extremely simple conditions can be stated for the following cases: H is a (kg, kf)-graph obtained from G by duplicating edges (Th. 1); H is a (kg, kf)-graph obtained from G by duplicating edges and by removing edges (Th. 2); H is a (2g, 2f)-graph obtained from G by duplicating and removing edges that contains an arbitrary edge ((Th. 4); H is a (1, f)-graph obtained from G by removing edges (Th. 5); H is a graph obtained from G by removing edges so that the maximum degree and the minimum degree have a ratio less than k (Th. 5); H is a (1, f)-graph obtained from G by removing edges and containing an arbitrary edge (Th. 7).

Optimal packings of edge-disjoint odd cycles

Abstract In a simple graph, we consider the minimum number of edges which hit all the odd cycles and the maximum number of edge-disjoint odd cycles. When these two coefficients are equal, interesting questions can be posed. Related problems, but interchanging ‘ vertex-disjoint ’ and ‘ edge-disjoint ’, have been studied by Berge and Fouquet (Discrete Math. 169 (1997) 169–176.)

Combinatorial games on a graph

Survey of various problems about combinatorial games.

On the optimal transversals of the odd cycles

Let G be a simple graph and let X be its vertex-set. A set T ⊆ X is a transversal of the odd cycles if it meets all the odd cycles of G. Let HG denote the family of the odd cycles of G (as subsets of X) which are chordless, i.e. minimal relatively to inclusion. Clearly, the minimum cardinality of a transversal T is the transversal number of the hypergraph HG, that is, with the notations of Hypergraph Theory (see [2]), min|T|=τ(HG) In this paper, we study the coefficient τ(HG); an unsolved problem is: For which graph G is this coefficient equal to the maximum number of pairwise disjoint odd cycles (the ‘Konig Property’)?

Abstract: Algorithms and Extremal Problems for Equipartite Colorings in Graphs and Hypergraphs

Publisher Summary This chapter describes algorithms and extremal problems for equipartite colorings in graphs and hypergraphs. The existence of an equipartite k -coloring for the edges of a graph is known for k larger than or equal to the chromatic index; a simple algorithm can be obtained from this theorem (Mc Diarmid, de Werra). The existence of an equipartite k -coloring for the vertices of a graph has been proved by Hajnal and Szemeredi for k larger than or equal to the maximum degree plus one; in this case, an algorithm can also be obtained. The structure of all graphs of order n with no equipartite k -coloring and having a minimum number of edges have been characterized in cooperation with Sterboul.