342 research outputs found

    Hamilton cycles in graphs and hypergraphs: an extremal perspective

    Full text link
    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

    Embedding large subgraphs into dense graphs

    Full text link
    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

    Hamilton decompositions of regular expanders: applications

    Get PDF
    In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our result on robust outexpanders; (ii) best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range of values for d. (iii) a similar result for digraphs of given minimum semidegree; (iv) an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; (v) the observation that dense quasi-random graphs are robust outexpanders; (vi) a verification of the `very dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.Comment: final version, to appear in J. Combinatorial Theory

    Optimal Hamilton covers and linear arboricity for random graphs

    Full text link
    In his seminal 1976 paper, P\'osa showed that for all pClogn/np\geq C\log n/n, the binomial random graph G(n,p)G(n,p) is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of G(n,p)G(n,p) with Hamilton cycles? How many cycles are necessary? In this paper we show that for pClogn/n p\geq C\log n/n, we can cover GG(n,p)G\sim G(n,p) with precisely Δ(G)/2\lceil\Delta(G)/2\rceil Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability pp, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.Comment: 13 page

    A bandwidth theorem for approximate decompositions

    Get PDF
    We provide a degree condition on a regular nn-vertex graph GG which ensures the existence of a near optimal packing of any family H\mathcal H of bounded degree nn-vertex kk-chromatic separable graphs into GG. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk\delta_k be the infimum over all δ1/2\delta\ge 1/2 ensuring an approximate KkK_k-decomposition of any sufficiently large regular nn-vertex graph GG of degree at least δn\delta n. Now suppose that GG is an nn-vertex graph which is close to rr-regular for some r(δk+o(1))nr \ge (\delta_k+o(1))n and suppose that H1,,HtH_1,\dots,H_t is a sequence of bounded degree nn-vertex kk-chromatic separable graphs with ie(Hi)(1o(1))e(G)\sum_i e(H_i) \le (1-o(1))e(G). We show that there is an edge-disjoint packing of H1,,HtH_1,\dots,H_t into GG. If the HiH_i are bipartite, then r(1/2+o(1))nr\geq (1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs GG of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.Comment: Final version, to appear in the Proceedings of the London Mathematical Societ

    Optimal path and cycle decompositions of dense quasirandom graphs

    Get PDF
    Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<10<p<1 be constant and let GGn,pG\sim G_{n,p}. Let odd(G)odd(G) be the number of odd degree vertices in GG. Then a.a.s. the following hold: (i) GG can be decomposed into Δ(G)/2\lfloor\Delta(G)/2\rfloor cycles and a matching of size odd(G)/2odd(G)/2. (ii) GG can be decomposed into max{odd(G)/2,Δ(G)/2}\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\} paths. (iii) GG can be decomposed into Δ(G)/2\lceil\Delta(G)/2\rceil linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
    corecore