56,173 research outputs found

    Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

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    For an increasing monotone graph property \mP the \emph{local resilience} of a graph GG with respect to \mP is the minimal rr for which there exists of a subgraph HGH\subseteq G with all degrees at most rr such that the removal of the edges of HH from GG creates a graph that does not possesses \mP. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model \GNP and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random dd-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive ϵ\epsilon and large enough values of dd with high probability the local resilience of the random dd-regular graph, \GND, with respect to being Hamiltonian is at least (1ϵ)d/6(1-\epsilon)d/6. We also prove that for the Binomial random graph model \GNP, for every positive ϵ>0\epsilon>0 and large enough values of KK, if p>Klnnnp>\frac{K\ln n}{n} then with high probability the local resilience of \GNP with respect to being Hamiltonian is at least (1ϵ)np/6(1-\epsilon)np/6. Finally, we apply similar techniques to Positional Games and prove that if dd is large enough then with high probability a typical random dd-regular graph GG is such that in the unbiased Maker-Breaker game played on the edges of GG, Maker has a winning strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur

    Local resilience of an almost spanning kk-cycle in random graphs

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    The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any k2k \geq 2, every graph on nn vertices with minimum degree kn/(k+1)kn/(k + 1) contains the kk-th power of a Hamilton cycle. We extend this result to a sparse random setting. We show that for every k2k \geq 2 there exists C>0C > 0 such that if pC(logn/n)1/kp \geq C(\log n/n)^{1/k} then w.h.p. every subgraph of a random graph Gn,pG_{n, p} with minimum degree at least (k/(k+1)+o(1))np(k/(k + 1) + o(1))np, contains the kk-th power of a cycle on at least (1o(1))n(1 - o(1))n vertices, improving upon the recent results of Noever and Steger for k=2k = 2, as well as Allen et al. for k3k \geq 3. Our result is almost best possible in three ways: for pn1/kp \ll n^{-1/k} the random graph Gn,pG_{n, p} w.h.p. does not contain the kk-th power of any long cycle; there exist subgraphs of Gn,pG_{n, p} with minimum degree (k/(k+1)+o(1))np(k/(k + 1) + o(1))np and Ω(p2)\Omega(p^{-2}) vertices not belonging to triangles; there exist subgraphs of Gn,pG_{n, p} with minimum degree (k/(k+1)o(1))np(k/(k + 1) - o(1))np which do not contain the kk-th power of a cycle on (1o(1))n(1 - o(1))n vertices.Comment: 24 pages; small updates to the paper after anonymous reviewers' report

    Local resilience for squares of almost spanning cycles in sparse random graphs

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    In 1962, P\'osa conjectured that a graph G=(V,E)G=(V, E) contains a square of a Hamiltonian cycle if δ(G)2n/3\delta(G)\ge 2n/3. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every ϵ>0\epsilon > 0 and p=n1/2+ϵp=n^{-1/2+\epsilon} a.a.s. every subgraph of Gn,pG_{n,p} with minimum degree at least (2/3+ϵ)np(2/3+\epsilon)np contains the square of a cycle on (1o(1))n(1-o(1))n vertices. This is almost best possible in three ways: (1) for pn1/2p\ll n^{-1/2} the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for c<2/3c<2/3 a.a.s. Gn,pG_{n,p} contains a subgraph with minimum degree at least cnpcnp which does not contain the square of a path on (1/3+c)n(1/3+c)n vertices

    Packing spanning graphs from separable families

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    Let G\mathcal G be a separable family of graphs. Then for all positive constants ϵ\epsilon and Δ\Delta and for every sufficiently large integer nn, every sequence G1,,GtGG_1,\dotsc,G_t\in\mathcal G of graphs of order nn and maximum degree at most Δ\Delta such that e(G1)++e(Gt)(1ϵ)(n2)e(G_1)+\dotsb+e(G_t) \leq (1-\epsilon)\binom{n}{2} packs into KnK_n. This improves results of B\"ottcher, Hladk\'y, Piguet, and Taraz when G\mathcal G is the class of trees and of Messuti, R\"odl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees have maximum degree at most Δ\Delta. The proof uses the local resilience of random graphs and a special multi-stage packing procedure

    Local resilience of spanning subgraphs in sparse random graphs

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    For each real γ>0γ>0 and integers Δ≥2Δ≥2 and k≥1k≥1, we prove that there exist constants β>0β>0 and C>0C>0 such that for all p≥C(log⁡n/n)1/Δp≥C(log⁡n/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most βn and at least Cmax⁡{p−2,p−1log⁡n}Cmax⁡{p−2,p−1log⁡n} vertices not in triangles

    Generating random graphs in biased Maker-Breaker games

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    We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b=o(n)b=o\left(\sqrt{n}\right), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1:b)(1:b) game on E(Kn)E(K_n). As another application, we show that for b=Θ(n/lnn)b=\Theta\left(n/\ln n\right), playing a (1:b)(1:b) game on E(Kn)E(K_n), Maker can build a graph which contains copies of all spanning trees having maximum degree Δ=O(1)\Delta=O(1) with a bare path of linear length (a bare path in a tree TT is a path with all interior vertices of degree exactly two in TT)
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