Robert Cori

On the Sandpile Group of Dual Graphs

The group of recurrent configurations in the sandpile model, introduced by Dhar , may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The aim of this paper is to prove that the sandpile group of planar graph is isomorphic to that of its dual. A combinatorial point of view on the subject is also developed.

Recognizable subsets of some partially abelian monoids

We show that, in a free partially abelian monoid generated by a finite alphabet A, the subset [X∗] of A∗ containing all the words equivalent to a product of words of X is rational if X is a finite set of words, each word containing at least one occurrence of any letter of A. We suppose that the graph the vertices of which are letters of A and the edges of which correspond to noncommuting pairs of letters is connected.

The sand-pile model and Tutte polynomials

A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given. This mapping is such that the difference between the number of gains on a configuration and the external activity of the associate tree is the number of edges of the graph. It gives a bijective proof of a result of Merino Lopez expressing the generating function of recurrent configurations as an evaluation of the Tutte polynomial.

Shuffle of parenthesis systems and Baxter permutations

Abstract A formula for the number alternating Baxter permutations is given. The proof of this formula is given by constructing bijection between permutations, trees, and words. This gives also a combinatorial proof of a formula appearing in the enumerative theory of planar maps.

Counting non-isomorphic chord diagrams

Different formulas counting families of non-isomorphic chord diagrams are given: planar and toroidal ones and those of maximal genus. These formulas are obtained establishing results on the structure of the automorphism group of diagrams of a given genus.

On the Automorphism Group of a Planar Hypermap

Several authors [1,9] have investigated the problem of determining which groups can arise as automorphism groups of planar maps. The methods mostly used are topological in nature and rest upon a theorem of Kerekjarto [8] and one of Eilenberg [4] which states that a periodic homeomorphism of the sphere has exactly two fixed points. Taking any map embedded in the sphere, one can extend any automorphism of it in order to obtain a homeomorphism of the sphere. An application of the above result yields

Some combinatorial aspects of time-stamp systems

Abstract The aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2k - 1 and an effective construction has been given which needs k2k - 1 vertices. This problems is issued from some questions related to the labeling of processes in a system for determining the order in which they were created. By modifying some requirements on the distributed system, we show that there arise other combinatorial structures leading to the construction of solutions the size of which becomes a linear function of the input.

Approximation of a TRace, Asynchronous Automata and the Ordering of Events in a Distributed System

The notion of trace was introduced in order to modelize the concurrency of actions. A trace is an element of the quotient of the free monoid by the congruence generated by a finite set of relations of the form ab∼ba. We introduce the notion of the approximation of a trace and we study its properties. The main result is the existence of an asynchronous automaton which recognizes the set of traces corresponding to an approximation. We give two applications : the first one is a new proof of Zielonkas theorem, and the second one is an algorithm for the ordering of events in a distributed system.

Construction of maps with prescribed automorphism group

Abstract We show that given a finite group G a hypermap can be constructed having automorphism group isomorphic to G. The bipartite map of a modified version of this hypermap then affords a map with the same property.

The Complexity of A Planar Hypermap And That of Its Dual

The number of spanning trees in a graph is often called its complexity [1]. A tree is of course of complexity one and it is a classical result of Cayley that the complete graph Kn has complexity nn-2. Between these two numbers lies the complexity of a connected graph. In the case of planar maps it is well-known that a map and its dual have equal complexity (see for instance [2]). in this communication we show that a similar result holds for hypermaps. To prove this result we use a diagram containing six hypermaps which is very related to W.T. Tuttes Trinity [9]. All the needed definitions are recalled. We thus give a few details about hypermaps, their underlying hypergraphs and the maps associated to them. The definition of a spanning hypertree is also given.