Charles Payan

Group connectivity of graphs: a nonhomogeneous analogue of nowhere-zero flow properties

Abstract Let G = (V, E) be a digraph and f a mapping from E into an Abelian group A. Associated with f is its boundary ∂f, a mapping from V to A, defined by ∂f(x) = Σe leaving xf(e)−Σe entering xf(e). We say that G is A-connected if for every b: V → A with Σx ∈ Vb(x)=0 there is an f: E → A − {0} with b = ∂f. This concept is closely related to the theory of nowhere-zero flows and is being studied here in light of that theory.

On the chromatic number of cube-like graphs

Abstract A cube-like graph is a graph whose vertices are all 2 n subsets of a set E of cardinality n , in which two vertices are adjacent if their symmetric difference is a member of a given specified collection of subsets of E . Many authors were interested in the chromatic number of such graphs and thought it was always a power of 2. Although this conjecture is false (we show a cube-like graph of chromatic number 7), we prove that there is no cube-like graph with chromatic number 3.

A class of threshold and domishold graphs: equistable and equidominating graphs

A threshold graph (respectively domishold graph) is a graph for which the independent sets (respectively the dominating sets) can be characterized by the 0, 1-solutions of a linear Inequality (see [1] and [3]). We define here the graphs for which the maximal independent sets (respectively the minimal dominating sets) are characterized by the 0, 1-solutions of a linear equation. Such graphs are said to be equistable (respectively equldominating). We characterize (by their architectural structure and by forbidden induced subgraphs) threshold graphs and domishold graphs which are equistable or equidominating. A larger class of equistable graphs is also presented.

Some progress in the packing of equal circles in a square

The problem of the densest packing of n equal circles in a square has been solved for n<10 in [4, 6]; and some solutions have been proposed for n ⩾ 10. In this paper we give some better packings for n = 10, 11, 13 and 14.

Upper embeddability and connectivity of graphs

In this paper, we prove that every cyclically 4-edge-connected graph is upper-embeddable. The proof is based on the study of the effect on the maximum genus of a graph when one of its subgraphs is collapsed to a vertex.

Variations on Tilings in the Manhattan Metric

We investigate tilings of the integer lattice in the Euclidean n-dimensional space. The tiles considered here are the union of spheres defined by the Manhattan metric. We give a necessary condition for the existence of such a tiling for Zn when n ≥ 2. We prove that this condition is sufficient when n=2. Finally, we give some tilings of Zn when n ≥ 3.

Domination number of the cross product of paths

Abstract The problem of determining the domination number of an arbitrary grid graph is known to be NP-complete, but the complexity of the same problem on complete grid graphs is still unknown. In the present paper we study the same problem on a similar grid graph defined by the cross product of two paths pk and Pn.

Flips Signés et Triangulations d'un Polygone

In this paper, we prove that any two triangulations of a given polygon may be transformed into one another by a signable sequence of flips if and only if every planar graph is 4-colorable. This result prove a conjecture due to Eliahou. Dans ce papier, on montre que l?on peut passer de toute triangulation d?un polygone a tout autre triangulation par application de flips signes si et seulement si on peut colorier tout graphe planaire en 4 couleurs. Ce resultat prouve une conjecture de Eliahou.

A class of upper‐embeddable graphs

In this paper, we prove the following result: Every graph obtained by connecting (with any number of edges) two vertex-disjoint upper-embeddable graphs graphs with even Betti number is upper-embeddable.

Arbres avec un nombre maximum de sommets pendants

This article is an attempt to study the following problem: Given a connected graph G, what is the maximum number of vertices of degree 1 of a spanning tree of G? For cubic graphs with n vertices, we prove that this number is bounded by 14n and 12(n - 2).