63,097 research outputs found

    Cycle packing

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    In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear minimum degree.Comment: 18 page

    Counting Hamilton cycles in sparse random directed graphs

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    Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if p(logn+ω(1))/np\ge(\log n+\omega(1))/n then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically n!(p(1+o(1)))nn!(p(1+o(1)))^{n}. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically n!(logn/n(1+o(1)))nn!(\log n/n(1+o(1)))^{n} directed Hamilton cycles

    Finding long cycles in graphs

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    We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three

    Edge-disjoint Hamilton cycles in graphs

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    In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2+α)n(1/2 + \alpha)n can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio

    Hamilton decompositions of regular tournaments

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    We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \eta>0 every regular tournament G of sufficiently large order n contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our main result. To appear in the Proceedings of the LM
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