88 research outputs found

    Cycle factors and renewal theory

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    For which values of kk does a uniformly chosen 33-regular graph GG on nn vertices typically contain n/k n/k vertex-disjoint kk-cycles (a kk-cycle factor)? To date, this has been answered for k=nk=n and for kâ‰Șlog⁥nk \ll \log n; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on kk-cycles. Here we settle the problem completely: the threshold for a kk-cycle factor in GG as above is Îș0log⁥2n\kappa_0 \log_2 n with Îș0=[1−12log⁥23]−1≈4.82\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82. Precisely, we prove a 2-point concentration result: if k≄Îș0log⁥2(2n/e)k \geq \kappa_0 \log_2(2n/e) divides nn then GG contains a kk-cycle factor w.h.p., whereas if k<Îș0log⁥2(2n/e)−log⁥2nnk<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph G(n,p)G(n,p). The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page

    Pancyclicity of 4-connected {Claw, Generalized Bull}-free Graphs

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    A graph G is pancyclic if it contains cycles of each length ℓ, 3 ≀ ℓ ≀ |V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths P i+1 and P j+1 with distinct vertices of a triangle. Gould, Luczak and Pfende

    Proof of Koml\'os's conjecture on Hamiltonian subsets

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    Koml\'os conjectured in 1981 that among all graphs with minimum degree at least dd, the complete graph Kd+1K_{d+1} minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when dd is sufficiently large. In fact we prove a stronger result: for large dd, any graph GG with average degree at least dd contains almost twice as many Hamiltonian subsets as Kd+1K_{d+1}, unless GG is isomorphic to Kd+1K_{d+1} or a certain other graph which we specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ

    Ramsey properties of randomly perturbed graphs: cliques and cycles

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    Given graphs H1,H2H_1,H_2, a graph GG is (H1,H2)(H_1,H_2)-Ramsey if for every colouring of the edges of GG with red and blue, there is a red copy of H1H_1 or a blue copy of H2H_2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3,Kt)(K_3,K_t)-Ramsey (for t≄3t\ge 3). They also raised the question of generalising this result to pairs of graphs other than (K3,Kt)(K_3,K_t). We make significant progress on this question, giving a precise solution in the case when H1=KsH_1=K_s and H2=KtH_2=K_t where s,t≄5s,t \ge 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3,Kt)(K_3,K_t)-Ramsey question. Moreover, we give bounds for the corresponding (K4,Kt)(K_4,K_t)-Ramsey question; together with a construction of Powierski this resolves the (K4,K4)(K_4,K_4)-Ramsey problem. We also give a precise solution to the analogous question in the case when both H1=CsH_1=C_s and H2=CtH_2=C_t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs,Kt)(C_s,K_t)-Ramsey (for odd s≄5s\ge 5 and t≄4t\ge 4).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil Powierski, stated results for cliques in graphs of low positive density separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to appear in CP

    A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length

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    International audienceWe describe a general purpose algorithm for counting simple cycles and simple paths of any length on a (weighted di)graph on N vertices and M edges, achieving a time complexity of O N + M + ω + ∆ |S |. In this expression, |S | is the number of (weakly) connected induced subgraphs of G on at most vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where (ω−1 ∆ −1 +1)|S | ≀ |Cycle |, with |Cycle | the total number of simple cycles of length at most , including backtracks and self-loops. On ErdƑs-RĂ©nyi random graphs, we find empirically that this happens when the edge probability is larger than circa 4/N. In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on n letters with a time complexity of O N + M + n ω + ∆ |S |. A Matlab implementation of the algorithm proposed here is available for download

    Pancyclicity of highly connected graphs

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    A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph GG satisfies Îș(G)≄α(G)\kappa(G) \ge \alpha(G), where Îș(G)\kappa(G) is the vertex-connectivity of GG, then GG has a Hamilton cycle. We prove a similar result implying that a graph GG is pancyclic, namely it contains cycles of all lengths between 33 and ∣G∣|G|: if ∣G∣|G| is large and Îș(G)>α(G)\kappa(G) > \alpha(G), then GG is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs, and improves upon a very recent result of Dragani\'c, Munh\'a-Correia, and Sudakov.Comment: 31 pages, 11 figure

    Mean quantum percolation

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    We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the twodimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erd'os-Rényi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum. © European Mathematical Society 2017

    On prisms, M\"obius ladders and the cycle space of dense graphs

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    For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main purpose of this paper is to prove the following: for every s > 0 there exists n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all circuits of X having length either f_0(X)-1 or f_0(X) generates all of Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure

    Combinatorics, Probability and Computing

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    One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly eïŹ€ective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
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