Balanced spanning trees in random geometric graphs
In a recent breakthrough, Montgomery showed that the Erdos-Rényi random graph G(n,p) typically contains all n-vertex trees of maximum degree Delta slightly above the (sharp) connectivity threshold. We consider the random geometric graph G_d(n,r) obtained by independently assigning a uniformly random position in [0,1]^d ...
Edge k-q-colorability of graphs
summary: Given positive integers k, q, we say that a graph is edge k-q-colorable if its edges can be colored in such a way that the number of colors incident to each vertex is at most q and that the size of a largest color class is at most k ...
The freezing threshold for uniformly random colourings of sparse graphs
summary: Given a random Δ-regular graph G, it holds that χ(G) ~ Δ / (2 ln Δ) with high probability. However, for any ε > 0 and k large enough, no (randomised) polynomial-time algorithm returning a proper k-colouring of such a random graph G is known to exist when k < (1−ε ...
Polytopes : théorèmes importants et généralisations conjecturales sur des espaces tropicaux
Certains théorèmes importants concernant les polytopes, par exemple le g-théorème et le problème de Minkowski, se trouvent à l'interaction de domaines des mathématiques dont on ne soupçonnerait pas la diversité au premier abord. Divers résultats plus ou moins récents en géométrie algébrique, en géométrie tropicale ou en théorie de ...
Musical juggling: from combinatorial modeling to computer assistance with artistic creation (a thesis in the making)
summary: Musical juggling consists of producing music by the very act of juggling. In this context (stemming from a
collaboration between computer science researchers at LISN and a juggling artist), we refer more specifically to
juggling with balls that each produce a musical note when caught.
The aim of this ...
The χ-binding function of d-directional segment graphs
summary: To color a graph properly, one needs at least as many colors as the size of its biggest clique; therefore, the chromatic number χ is lower-bounded by the clique number ω. In general, there are no upper bounds on χ in terms of ω. The graphs for which we ...
Complexité en états : Renverser un langage réduit la complexité de l'opération racine.
summary: Les automates (DFA) sont des machines à états qui acceptent ou rejettent des mots. L'ensemble des mots reconnus par un automate est son langage. Les langages rationnels coïncident avec les langages reconnaissable par des automates. Ici nous allons nous intéresser à une mesure, à savoir la complexité en ...
Forecasting multivariate time series with attention mechanism and unsupervised learning
summary: In the realm of newborn healthcare, identifying neurological pathologies has traditionally relied on the expertise of medical professionals, who perform visual assessments. However, due to the limited number of such experts available, there is an urgent need to develop a pre-diagnostic tool capable of early detection of abnormal neurological ...
Réalisation de l'algèbre d'Okada et correspondance de Robinson-Schensted-Fomin du treillis de Young-Fibonacci
Il est bien connu que le treillis de Young peut s'interpréter comme le diagramme de Bratelli des groupes symétriques, décrivant, par exemple, comment les représentations irréductibles se restreignent de Sn à S_n-1. En 1975, Stanley a découvert un treillis similaire appelée treillis de Young-Fibonacci qui a été interprété comme ...
Beyond the fractional Reed bound for triangle-free graphs
summary: The notion of fractional colouring is an important concept in graph theory that is commonly used to extend the notion of graph colouring beyond integer values. It is a relaxation of the traditional chromatic number, allowing for real-valued weights or probabilities associated with each independent set of a graph ...