406 research outputs found

    Combinatorial theorems relative to a random set

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    We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

    Embedding large subgraphs into dense graphs

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    What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

    Ramsey numbers of ordered graphs

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    An ordered graph is a pair G=(G,)\mathcal{G}=(G,\prec) where GG is a graph and \prec is a total ordering of its vertices. The ordered Ramsey number R(G)\overline{R}(\mathcal{G}) is the minimum number NN such that every ordered complete graph with NN vertices and with edges colored by two colors contains a monochromatic copy of G\mathcal{G}. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings Mn\mathcal{M}_n on nn vertices for which R(Mn)\overline{R}(\mathcal{M}_n) is superpolynomial in nn. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number R(G)\overline{R}(\mathcal{G}) is polynomial in the number of vertices of G\mathcal{G} if the bandwidth of G\mathcal{G} is constant or if G\mathcal{G} is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

    Bandwidth theorem for random graphs

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    A graph GG is said to have \textit{bandwidth} at most bb, if there exists a labeling of the vertices by 1,2,...,n1,2,..., n, so that ijb|i - j| \leq b whenever {i,j}\{i,j\} is an edge of GG. Recently, B\"{o}ttcher, Schacht, and Taraz verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every positive r,Δ,γr,\Delta,\gamma, there exists β\beta such that if HH is an nn-vertex rr-chromatic graph with maximum degree at most Δ\Delta which has bandwidth at most βn\beta n, then any graph GG on nn vertices with minimum degree at least (11/r+γ)n(1 - 1/r + \gamma)n contains a copy of HH for large enough nn. In this paper, we extend this theorem to dense random graphs. For bipartite HH, this answers an open question of B\"{o}ttcher, Kohayakawa, and Taraz. It appears that for non-bipartite HH the direct extension is not possible, and one needs in addition that some vertices of HH have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed rr-chromatic graph H0H_0 which one can find in a spanning subgraph of G(n,p)G(n,p) with minimum degree (11/r+γ)np(1-1/r + \gamma)np.Comment: 29 pages, 3 figure

    On deficiency problems for graphs

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    Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property P\mathcal P and a graph GG, the deficiency def(G)\text{def}(G) of the graph GG with respect to the property P\mathcal P is the smallest non-negative integer tt such that the join GKtG*K_t has property P\mathcal P. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an nn-vertex graph GG needs to ensure GKtG*K_t contains a KrK_r-factor (for any fixed r3r\geq 3). In this paper we resolve their problem fully. We also give an analogous result which forces GKtG*K_t to contain any fixed bipartite (n+t)(n+t)-vertex graph of bounded degree and small bandwidth.Comment: 11 page

    Hamilton cycles in graphs and hypergraphs: an extremal perspective

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    As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

    Tilings in randomly perturbed dense graphs

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    A perfect HH-tiling in a graph GG is a collection of vertex-disjoint copies of a graph HH in GG that together cover all the vertices in GG. In this paper we investigate perfect HH-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds mm random edges to it. Specifically, for any fixed graph HH, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect HH-tiling with high probability. Our proof utilises Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect HH-tilings in dense graphs.Comment: 19 pages, to appear in CP
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