Labeled Packing of Cycles and Circuits

In 2013, Duch{\^e}ne, Kheddouci, Nowakowski and Tahraoui [4, 9] introduced a labeled version of the graph packing problem. It led to the introduction of a new parameter for graphs, the k-labeled packing number $\lambda$ k. This parameter corresponds to the maximum number of labels we can assign to the vertices of the graph, such that we will be able to create a packing of k copies of the graph, while conserving the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle C n of order n = 2k + x, with k $\ge$ 2 and 1 $\le$ x $\le$ 2k -- 1, the value of $\lambda$ k (C n) was 2 if x was 1 and k was even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counter example. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits

A proof of Ringel's Conjecture

A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.Comment: 37 pages, 4 figure

Embedding rainbow trees with applications to graph labelling and decomposition

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally k-bounded if each vertex is contained in at most k edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph Kn contains a rainbow copy of every tree with at most (1−o(1))n/k vertices. As a locally k-bounded edge-colouring of Kn may have only (n−1)/k distinct colours, this is essentially tight. As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory. Firstly, we prove an asymptotic version of Ringel's conjecture from 1963, showing that any n-edge tree packs into the complete graph K(2n+o(n)) to cover all but o(n^2) of its edges. Secondly, we show that all trees have an almost-harmonious labelling. The existence of such a labelling was conjectured by Graham and Sloane in 1980. We also discuss some additional applications

A proof of Ringel’s conjecture

A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs 2n+1 times into the complete graph K2n+1. In this paper, we prove this conjecture for large n

HABILITATION A DIRIGER DES RECHERCHES Graphes et jeux combinatoires

On considère généralement que la théorie des graphes est née au 18e siècle, et qu'elle connaît un essor significatif depuis les années 1960. L'avènement de la théorie des jeux combinatoires est quant à lui plus récent (fin des années 1970). Ce domaine reste alors moins exploré dans la littérature, et de nombreuses études sur des techniques générales de résolution sont toujours actuellement en cours de construction. Dans ce mémoire, je propose plusieurs tours d'horizons à propos de problématiques bien ciblées de ces deux domaines.Dans un premier temps, je m'interroge sur la complexité des règles de jeux de suppression de tas. Il s'avère que dans la littérature, la complexité d'un jeu est souvent définie comme la complexité algorithmique d'une stratégie gagnante. Cependant, il peut aussi avoir du sens de s'interroger sur la nature des règles de jeu. Un premier pas dans cette direction a été fait avec l'introduction du concept de jeu dit invariant. On notera au passage que certains résultats obtenus ont mis en exergue des liens entre combinatoire des mots et stratégie gagnante d'un jeu. Dans un deuxième chapitre, j'aborde les jeux sous l'angle des graphes. Deux aspects sont considérés:* Un graphe peut être vu comme un support de jeu. Le cas du jeu de Nim et ses variantes sur les graphes y est examiné.* Certaines problématiques standard de théorie des graphes peuvent être transformées dans une version ludique. C'est d'ailleurs un objet d'étude de plus en plus prisé par la communauté. Nous détaillerons le cas des jeux de coloration sommet.Enfin, le dernier chapitre se concentre sur deux nouvelles variantes de problématiques issues de la théorie des graphes: le placement de graphes et les colorations distinguantes. J'en profite pour faire un état de l'art des principaux résultats sur ces deux domaines

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..