Henry Meyniel

On the perfect graph conjecture

We prove the following theorem: If every, odd cycle of length ≥5 has at least two chords, then the graph is perfect. This generalizes a result of Gallai and Suranyi and also a result of Olaru and Sachs.

On a game of policemen and robber

Abstract We consider a game where policemen try to catch a robber on a graph G (as defined by A. Quilliot) and we find the exact minimal number of policemen needed when G is a Cartesian product of trees.

On a problem of G. Hahn about coloured hamiltonian paths in K2t

Abstract We prove a theorem showing that for every integer p (p⩾2) there is a good minimal coloring of the edges of K2p such that every hamiltonian path in K2p uses at least one colour twice. This gives a counter-example to a conjecture of Hahn [2].

About quasi-kernels in a digraph

Abstract We generalize a result by Maghout who had shown that every tournament of radius 2 admits three distinct centers. Here we prove that every graph without kernel has at least three distinct quasi-kernels.

Kernels in directed graphs: a poison game

Abstract We consider a two player game on a progressively and locally finite directed graph and we prove that the first player wins if and only if the graph has a local kernel. The result is sharp. From it, we derive a short proof of a general version of the Galeana-Sanchez & Neuman-Lara Theorem that give a sufficient condition for a digraph to be kernel-perfect.

On a relationship between Hadwiger and stability numbers

Abstract In [2] it is proved that the inequality η ( G )·(2 α ( G ) − 1⩾ n ( G ) holds for any graph G where η ( G ) denotes the Hadwiger number of G , α(G) its stability number and n ( G ) its number of vertices, and it was conjectured the inequality η ( G )· α ( G ) ⩾ n ( G ) holds for every graph G . In this note, the graphs satisfying the equality case of the above mentioned theorem are characterized; an equivalent of the above conjecture is given and we define two parameters related to it and give their bound.

Some operations preserving the existence of kernels

Abstract We consider some classical constructions of graphs: join of two graphs, duplication of a vertex, and show that they behave nicely from the point of view of the existence of kernels.

The vertex picking game and a variation of the game of dots and boxes

Abstract A two-player game played on a graph is introduced and completely solved. As a consequence, a solution to a simplified variation of a well-known game called dots and boxes played on a grid [2] is given.

On quasi-kernels of a graphs

Abstract We define three classes of quasi-kernels for a directed graph. As a consequence, we show the existence of quasi-kernels in every progressively finite graph and in every locally finite graph, generalizing the result of Chvatal and Lovasz which deals with the finite case. Our method shows that the problem of finding a quasi-kernel in a finite digraph and the problem of finding the unique kernel of an acircuitic finite digraph have the same algorithmic complexity.We define three classes of quasi-kernels for a directed graph. As a consequence, we show the existence of quasi-kernels in every progressively finite graph and in every locally finite graph, generalizing the result of Chv~ital and Lov~tsz which deals with the finite case. Our method shows that the problem of finding a quasi-kernel in a finite digraph and the problem of finding the unique kernel of an acircuitic finite digraph have the same algorithmic complexity.

On the Shannon Capacity of a Directed Graph

The Shannon capacity problem in undirected graphs is well known [7]. This problem can be naturally generalized in the case of directed graphs and we find, in particular, the Shannon capacity of directed cycles of each length. For every graph, a lower and an upper bound are obtained.