On the threshold for the Maker‐Breaker <i>H</i>‐game

Nenadov, Rajko; Steger, Angelika; Stojaković, Miloš

doi:10.1002/rsa.20628

Cited by 22 publications

(38 citation statements)

References 18 publications

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“…Next, a moment's thought shows that if a graph is 2-bounded anti-Ramsey for F , then the Breaker has a winning strategy in the Maker-Breaker F -game. Thus our result implies the result of Nenadov, Steger and Stojaković [12], where the authors have characterized edge probabilities for which a random graph is Maker's win in the Maker-Breaker Fgame. Finally, for the anti-Ramsey problem in random graphs with respect to proper-colourings we determine the lower bound for all large enough cliques and cycles.…”

Section: Introductionsupporting

confidence: 72%

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

Nenadov¹,

Škorić²,

Steger³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability p to a deterministic question of whether there exists a finite graph that forms an obstruction. In the second part of the paper we apply this framework to address and solve various open problems. In particular, we provide a matching lower bound for the result of Friedgut, Rödl and Schacht (2010) and, independently, Conlon and Gowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. A problem that was open for more than 15 years. We also improve a result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs and extend it to hypergraphs. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, .

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Section: Introductionsupporting

confidence: 72%

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

Nenadov¹,

Škorić²,

Steger³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…On the random geometric graph we basically have the same witness of Maker's victory, as the smallest k for which Maker can win the triangle game on edges of K k is k = 5. Interestingly, for most other graphs H it is known that a hitting time result for Maker-win in the Erdős-Rényi random graph process cannot involve the appearance of a finite graph on which Maker can win-Maker-winning strategy must be of "global nature" [18,19]. This is in contrast to the results we obtained in Theorem 1.7, showing that on the random geometric graph Maker can typically win the H-game by simply spotting a copy of one of some finite list of graphs and restricting his attention to that subgraph.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…The limiting probability is known precisely only for several special classes of graphs [18]. Recently, the order of the threshold probability was determined for all graphs H that are not trees, and whose maximum 2-density is not determined by a K 3 as a subgraph [19]. When it comes to other models of random graphs, some positional games on random regular graphs were studied in [5].…”

Section: Introductionmentioning

confidence: 99%

Maker‐breaker games on random geometric graphs

Beveridge

Dudek

Frieze

et al. 2014

Random Struct Algorithms

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In a Maker-Breaker 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 Maker-Breaker 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 edge-length then, with probability tending to one as n → ∞, the graph becomes Maker-win 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 H-game 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 Maker-win 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.

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“…After the present work was submitted, Nenadov and Trujić [20] showed that Theorem 1.1 is true even if one replaces  k+1 by  2k+1 . Independently, a more general result was recently obtained by Antoniuk and the current authors [2].…”

Section: Note Added In Proofmentioning

confidence: 95%

Powers of Hamiltonian cycles in randomly augmented graphs

Dudek

Reiher

Ruciński

et al. 2019

Random Struct Algorithms

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We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every k ≥ 1 and sufficiently large n already the minimum degree δfalse(Gfalse)≥kk+1n for an n‐vertex graph G alone suffices to ensure the existence of a kth power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just O(n) random edges ensures the presence of the (k + 1)st power of a Hamiltonian cycle with probability close to one.

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On the threshold for the Maker‐Breaker H‐game

Cited by 22 publications

References 18 publications

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

Maker‐breaker games on random geometric graphs

Powers of Hamiltonian cycles in randomly augmented graphs

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