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13
On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs
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Hitting time results for MakerBreaker games
 IN PROCEEDINGS OF THE 22ND ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA’11
, 2011
"... We study MakerBreaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker’s win in a game in which Maker’s goal is to build a graph which admits some monotone increasing property P. We focus on three natural target ..."
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We study MakerBreaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker’s win in a game in which Maker’s goal is to build a graph which admits some monotone increasing property P. We focus on three natural target properties for Maker’s graph, namely being kvertexconnected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the kvertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. An extended abstract of this paper was previously published in [4].
Random Structures and Algorithms
, 2004
"... We provide an introduction to the analysis of random combinatorial structures and some of the associated computational problems. ..."
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We provide an introduction to the analysis of random combinatorial structures and some of the associated computational problems.
Long cycles in subgraphs of (pseudo)random directed graphs
 J. Graph Th
"... We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0 < γ < 1/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a l ..."
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We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0 < γ < 1/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G ′ be a subgraph of G with (1/2 + γ)E  edges. Then G ′ contains a directed cycle of length at least (c − o(1))n. Moreover, there is a subgraph G ′ ′ of G with (1/2 + γ − o(1))E edges that does not contain a cycle of length at least cn. 1
Corradi and Hajnal’s theorem for sparse random graphs
, 2010
"... In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n) 1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but a ..."
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In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n) 1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p −2) vertices. Moreover, the assumption on p is optimal up to the (log n) 1/2 factor and the presence of the set of O(p −2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3partite graph, then this graph contains a perfect triangle packing. 1
Random directed graphs are robustly Hamiltonian
, 2014
"... A classical theorem of GhouilaHouri from 1960 asserts that every directed graph on n vertices with minimum outdegree and indegree at least n/2 contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph D(n, p), that is, a directed graph in which every orde ..."
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A classical theorem of GhouilaHouri from 1960 asserts that every directed graph on n vertices with minimum outdegree and indegree at least n/2 contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph D(n, p), that is, a directed graph in which every ordered pair (u, v) becomes an arc with probability p independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if p log n/√n, then a.a.s. every subdigraph of D(n, p) with minimum outdegree and indegree at least (1/2 + o(1))np contains a directed Hamilton cycle. The constant 1/2 is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs.
Pancyclic subgraphs of random graphs
 J. Graph Theory
"... Abstract An nvertex graph is called pancyclic if it contains a cycle of length t for all 3 ≤ t ≤ n. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if p n −1/2 , then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamilton ..."
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Abstract An nvertex graph is called pancyclic if it contains a cycle of length t for all 3 ≤ t ≤ n. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if p n −1/2 , then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than ( 1 2 + o(1)) n 2 p edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion 1 2 of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich, Lee, and Sudakov. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers).
MakerBreaker games on random geometric graphs
, 2013
"... In a MakerBreaker game on a graph G, Breaker and Maker alternately claim edges of G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four MakerBreaker games played on random geometric graphs. For each of our four games we show ..."
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In a MakerBreaker game on a graph G, Breaker and Maker alternately claim edges of G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four MakerBreaker games played on random geometric graphs. For each of our four games we show that if we add edges between n points chosen uniformly at random in the unit square by order of increasing edgelength then, with probability tending to one as n → ∞, the graph becomes Makerwin the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the Hgame as soon as there is a subgraph from a finite list of “minimal graphs”. These results also allow us to give precise expressions for the limiting probability that G(n, r) is Makerwin in each case, where G(n, r) is the graph on n points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most r. 1