32,403 research outputs found

    Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five

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    In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on nn vertices and mm edges with girth at least four or five, respectively, has an induced forest on at least n29mn-\frac{2}{9}m or n15mn-\frac{1}{5}m vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most 29m\frac{2}{9}m or 15m\frac{1}{5}m, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2-connectivity requirement

    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

    Hamilton cycles in sparse robustly expanding digraphs

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    The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemer\'edi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to suitable sparse robustly expanding digraphs.Comment: Accepted for publication in The Electronic Journal of Combinatoric

    Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

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    A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show 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. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust decomposition lemma' for application in subsequent paper

    Monochromatic cycle covers in random graphs

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    A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of KnK_n with rr colors, there is a cover of its vertex set by at most f(r)=O(r2logr)f(r) = O(r^2 \log r) vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of KnK_n but only on the number of colors. We initiate the study of this phenomena in the case where KnK_n is replaced by the random graph G(n,p)\mathcal G(n,p). Given a fixed integer rr and p=p(n)n1/r+εp =p(n) \ge n^{-1/r + \varepsilon}, we show that with high probability the random graph GG(n,p)G \sim \mathcal G(n,p) has the property that for every rr-coloring of the edges of GG, there is a collection of f(r)=O(r8logr)f'(r) = O(r^8 \log r) monochromatic cycles covering all the vertices of GG. Our bound on pp is close to optimal in the following sense: if p(logn/n)1/rp\ll (\log n/n)^{1/r}, then with high probability there are colorings of GG(n,p)G\sim\mathcal G(n,p) such that the number of monochromatic cycles needed to cover all vertices of GG grows with nn.Comment: 24 pages, 1 figure (minor changes, added figure

    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

    Subset feedback vertex set is fixed parameter tractable

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    The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1
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