7 research outputs found

    The hamiltonicity and path t-coloring of Sierpiński-like graphs

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    AbstractA mapping ϕ from V(G) to {1,2,…,t} is called a path t-coloring of a graph G if each G[ϕ−1(i)], for 1≤i≤t, is a linear forest. The vertex linear arboricity of a graph G, denoted by vla(G), is the minimum t for which G has a path t-coloring. Graphs S[n,k] are obtained from the Sierpiński graphs S(n,k) by contracting all edges that lie in no induced Kk. In this paper, the hamiltonicity and path t-coloring of Sierpiński-like graphs S(n,k), S+(n,k), S++(n,k) and graphs S[n,k] are studied. In particular, it is obtained that vla(S(n,k))=vla(S[n,k])=⌈k/2⌉ for k≥2. Moreover, the numbers of edge disjoint Hamiltonian paths and Hamiltonian cycles in S(n,k), S+(n,k) and S++(n,k) are completely determined, respectively

    On the structure of graphs with forbidden induced substructures

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    One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every 22-edge-colouring of the complete graph on nn vertices there is a monochromatic clique on at least 12logn\frac{1}{2}\log n vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair (n,e)(n,e) is the family consisting of all graphs of order nn and size ee, i.e. on nn vertices with ee edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs (m,f)(m,f), i.e. for nn approaching infinity, the limit superior of the fraction of all possible sizes ee, such that the order-size pair (n,e)(n,e) does not avoid the pair (m,f)(m,f)

    Structures métriques et leurs groupes d’automorphismes : reconstruction, homogénéité, moyennabilité et continuité automatique

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    This thesis focuses on the study of Polish groups seen as automorphism groups of metric structures. The observation that every non-archimedean Polish group is the automorphism group of an ultrahomogeneous countable structure has indeed led to fruitful interactions between group theory and model theory. In the framework of metric model theory, introduced by Ben Yaacov, Henson and Usvyastov, this correspondence has been extended to all Polish groups by Melleray. In this thesis, we study various facets of this correspondence. The relationship between a structure and its automorphism group is particularly close in the setting of ℵ0-categorical structures. Indeed, the Ahlbrandt-Ziegler reconstruction theorem allows one to recover an ℵ0-categorical structure, up to bi-interpretability, from its automorphism group. In a joint work with Itai Ben Yaacov, we generalize this result to separably categorical metric structures. Besides, ultrahomogeneous countable structures have the advantage of being completely determined by their finitely generated substructures. In particular, this enabled Moore to give a combinatorial characterization of amenability for nonarchimedean Polish groups. We extend this characterization to all Polish groups and we deduce that amenability is a Gδ condition. Still in a reconstruction perspective, we are interested in the automatic continuity property for Polish groups. Sabok and Malicki introduced conditions of a combinatorial nature on an ultrahomogeneous metric structure that imply the automatic continuity property for its automorphism group. We show that these conditions carry to countable powers, which leads to the groups Aut(μ)N, U(l2)N and Iso(U)N satisfying the automatic continuity property. Those conditions are a weakening of the property of having ample generics. In a joint work with Francois Le Maitre, we exhibit the first examples of connected groups with ample generics, which answers a question of Kechris and Rosendal. Finally, in a joint work with Isabel Muller and Aristotelis Panagiotopoulos, we study the relative homogeneity of substructures in an ultrahomogeneous countable structure. We characterize it completely by a property of the types over the substructures: being determined by a finite setCette thèse porte sur l'étude des groupes polonais vus comme groupes d'automorphismes de structures métriques. L'observation que tout groupe polonais non archimédien est le groupe d'automorphismes d'une structure dénombrable ultra homogène a en effet mené à des interactions fructueuses entre la théorie des groupes et la théorie des modèles. Dans le cadre de la théorie des modèles métriques, introduite par Ben Yaacov, Henson et Usvyatsov, cette correspondance a été étendue par Melleray à tous les groupes polonais. Dans cette thèse, nous étudions diverses facettes de cette correspondance. Le lien entre une structure et son groupe d automorphismes est particulièrement étroit dans le cadre des structures ℵ0-categoriques. En effet, le théorème de reconstruction d'Ahlbrandt-Ziegler permet de retrouver une structure ℵ0-categorique, à bi-interprètabilité près, à partir de son groupe d'automorphismes. Dans un travail en commun avec Itai Ben Yaacov, nous généralisons ce résultat aux structures métriques separablement catégoriques. Les structures dénombrables ultra homogènes ont de plus l avantage d'être complètement déterminées par leurs sous-structures finiment engendrées. Cela a notamment permis a Moore de donner une caractérisation combinatoire de la moyennabilité des groupes polonais non archimédiens. Nous étendons cette caractérisation à tous les groupes polonais et nous en déduisons que la moyennabilite est une condition Gδ. Toujours dans une optique de reconstruction, nous nous intéressons à la propriété de continuité automatique pour les groupes polonais. Sabok et Malicki ont introduit des conditions de nature combinatoire sur une structure métrique ultra homogène qui impliquent la propriété de continuité automatique pour son groupe d'automorphismes. Nous montrons que ces conditions passent à la puissance dénombrable, ce qui a pour conséquence que les groupes Aut(μ)N, U(l2)N et Iso(U)N satisfont la propriété de continuité automatique. Ces conditions sont un affaiblissement du fait d'avoir des amples génériques. Dans un travail en commun avec Francois Le Maitre, nous exhibons les premiers exemples de groupes connexes qui ont des amples génériques, ce qui répond à une question de Kechris et Rosenda
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