9,250 research outputs found

    Edge-decompositions of graphs with high minimum degree

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    A fundamental theorem of Wilson states that, for every graph FF, every sufficiently large FF-divisible clique has an FF-decomposition. Here a graph GG is FF-divisible if e(F)e(F) divides e(G)e(G) and the greatest common divisor of the degrees of FF divides the greatest common divisor of the degrees of GG, and GG has an FF-decomposition if the edges of GG can be covered by edge-disjoint copies of FF. We extend this result to graphs GG which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3K_3-divisible graph of minimum degree at least 9n/10+o(n)9n/10+o(n) has a K3K_3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3K_3-divisible graph with minimum degree at least 3n/43n/4 has a K3K_3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3+o(n)2n/3 +o(n) for the existence of a C4C_4-decomposition, and of n/2+o(n)n/2+o(n) for the existence of a C2C_{2\ell}-decomposition, where 3\ell\ge 3. Our main contribution is a general `iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3K_3-divisible graph with minimum degree at least 3n/4+o(n)3n/4+o(n) has an approximate K3K_3-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement

    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

    Hamilton decompositions of regular expanders: applications

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    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

    Clique decompositions of multipartite graphs and completion of Latin squares

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    Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series

    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
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