3,014 research outputs found

    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

    Decomposing dense bipartite graphs into 4-cycles

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    Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles

    Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

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    We are motivated by a tantalizing open question in exact algorithms: can we detect whether an nn-vertex directed graph GG has a Hamiltonian cycle in time significantly less than 2n2^n? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0<λ<10<\lambda<1 and prime pp we can count the Hamiltonian cycles modulo p(1λ)n3pp^{\lfloor (1-\lambda)\frac{n}{3p}\rfloor} in expected time less than cnc^n for a constant c<2c<2 that depends only on pp and λ\lambda. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in O(3nα(G))O^*(3^{n-\alpha(G)}) time and polynomial space, where α(G)\alpha(G) is the size of the maximum independent set in GG. In particular, this yields an O(3n/2)O^*(3^{n/2}) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O(2k)O^*(2^k)-time randomized algorithm for detecting out-branchings with at least kk internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed kk-Leaf problem, based on a non-standard monomial detection problem

    Mod-phi convergence I: Normality zones and precise deviations

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    In this paper, we use the framework of mod-ϕ\phi convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables (Xn)nN(X_{n})_{n \in \mathbb{N}}, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations P[XntnB]P[X_{n} \in t_{n}B], instead of the usual estimates for the rate of exponential decay log(P[XntnB])\log( P[X_{n}\in t_{n}B]). Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

    Long path and cycle decompositions of even hypercubes

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    We consider edge decompositions of the nn-dimensional hypercube QnQ_n into isomorphic copies of a given graph HH. While a number of results are known about decomposing QnQ_n into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if nn is even, <2n\ell < 2^n and \ell divides the number of edges of QnQ_n, then the path of length \ell decomposes QnQ_n. Tapadia et al.\ proved that any path of length 2mn2^mn, where 2m<n2^m<n, satisfying these conditions decomposes QnQ_n. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to 2n+1/n2^{n+1}/n decompose QnQ_n. As a consequence, we show that QnQ_n can be decomposed into copies of any path of length at most 2n/n2^{n}/n dividing the number of edges of QnQ_n, thereby settling Erde's conjecture up to a linear factor
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