21 research outputs found

    k-Ordered Hamilton cycles in digraphs

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    Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, S\'ark\"ozy and Selkow

    The Directed Grid Theorem

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    The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed 97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f : N -> N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the Reed, Johnson, Robertson, Seymour and Thomas conjecture. Only very recently, this result has been extended to all classes of digraphs excluding a fixed undirected graph as a minor (see [Kawarabayashi, Kreutzer 14]). In this paper, nearly two decades after the conjecture was made, we are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality and to prove the directed grid theorem. As consequence of our results we are able to improve results in Reed et al. in 1996 [Reed, Robertson, Seymour, Thomas 96] (see also [Open Problem Garden]) on disjoint cycles of length at least l and in [Kawarabayashi, Kobayashi, Kreutzer 14] on quarter-integral disjoint paths. We expect many more algorithmic results to follow from the grid theorem.Comment: 43 pages, 21 figure

    Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs

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    We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))

    Half-integral Erd\H{o}s-P\'osa property of directed odd cycles

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    We prove that there exists a function f:NRf:\mathbb{N}\rightarrow \mathbb{R} such that every digraph GG contains either kk directed odd cycles where every vertex of GG is contained in at most two of them, or a vertex set XX of size at most f(k)f(k) hitting all directed odd cycles. This extends the half-integral Erd\H{o}s-P\'osa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.Comment: 16 pages, 5 figure

    Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

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    In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k,t) = O(k^7 t^4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists an integer h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t) = O(t^5) suffices.Comment: final version, to appear in Combinatoric

    Packings and coverings with Hamilton cycles and on-line Ramsey theory

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    A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let GG be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of GG with edge-disjoint Hamilton cycles. • Let TT be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of TT with edge-disjoint Hamilton cycles. • Let log 1^11^17^7 nn/nnpp≤1-nn^-1^1/^/8^8. We prove that GGn_n,_,p_p may a.a.s be covered by a set of ⌈Δ(GGn_n,_,p_p)/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r(GG,HH) denote the on-line Ramsey number of GG and HH. We conjecture the exact values of r (PPk_k,PP_ℓ) for all kk≤ℓ. We prove this conjecture for kk=2, prove it to within an additive error of 10 for kk=3, and prove an asymptotically tight lower bound for kk=4. We also determine r(PP3_3,CC_ℓ exactly for all ℓ

    A Polylogarithimic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

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    In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n-vertex graph G, a collection M={(s_1,t_1),...,(s_k,t_k)} of demand pairs and an integer c. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by c. When the maximum allowed congestion is c=1, this is the classical Edge-Disjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an O(n)O(\sqrt n)-approximation, by rounding the standard multi-commodity flow relaxation of the problem. This matches the Ω(n)\Omega(\sqrt n) lower bound on the integrality gap of this relaxation. We show an O(polylogk)O(poly log k)-approximation algorithm for EDPwC with congestion c=2, by rounding the same multi-commodity flow relaxation. This gives the best possible congestion for a sub-polynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of Ω~((logn)1/(c+1))\tilde{\Omega}((\log n)^{1/(c+1)}) for any constant congestion c. Prior to our work, the best approximation factor for EDPwC with congestion 2 was O~(n3/7)\tilde O(n^{3/7}), and the best algorithm achieving a polylogarithmic approximation required congestion 14
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