Building spanning trees quickly in Maker-Breaker games

Clemens, Dennis; Ferber, Asaf; Glebov, Roman; Hefetz, Dan; Liebenau, Anita

doi:10.1007/978-88-7642-475-5_58

Cited by 8 publications

(19 citation statements)

References 19 publications

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“…In view of (3), to complete the proof of Theorem 1.5 it suffices to show that a.a.s. Maker can ensure that f II (v) ≤ 9 10 εd G (v)p for all vertices v ∈ V (G) at the end of the game. Since Maker's goal here is to build a random graph, if a failure of type I occurs it does not harm Maker.…”

Section: Proof Of the Main Resultsmentioning

confidence: 70%

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Generating random graphs in biased Maker-Breaker games

Ferber

Krivelevich

Naves

2015

Random Struct. Alg.

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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), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1 : b) game on E(K n ). As another application, we show that for b = Θ (n/ ln n), playing a (1 : b) game on E(K n ), Maker can build a graph which contains copies of all spanning trees having maximum degree ∆ = O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T ).

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Section: Proof Of the Main Resultsmentioning

confidence: 70%

“…holds for all v ∈ V (G). To see this, recall that d G (v) ≥ δ(G) ≥ 10 ln n εp , so for any fixed v ∈ V (G), Lemma 2.1 implies that P[Bin(d G (v), p) < 9 10 d G (v)p] = o( 1 n ). Hence, by the union bound, a.a.s.…”

Section: Proof Of the Main Resultsmentioning

confidence: 92%

Generating random graphs in biased Maker-Breaker games

Ferber

Krivelevich

Naves

2015

Random Struct. Alg.

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“…Naturally, one may wonder whether the error term in the above theorem can be improved when we put stronger constraints on ∆(T ) and b. This was answered in [8] as follows. Theorem 1.10 (Theorem 1.1 and Theorem 1.4 in [8]).…”

Section: Introductionmentioning

confidence: 99%

Fast Strategies in Waiter-Client Games

Clemens

Gupta

Hamann

et al. 2020

Electron. J. Combin.

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Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.

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“…Even though in the literature this game does not draw as much interest as HAM n and HP n , it has appeared as an auxiliary game when studying some other games on graphs, see e.g. [2]. Note that in the game FHP n the edge between the fixed vertices u and v actually does not participate in any winning set, so right away we obtain the same result for the game played on E(K n ) \ {(u, v)}.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Maker–Breaker Games on Small Graphs

Stojaković

Trkulja

2019

Experimental Mathematics

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We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph K n , where Maker's goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for n = 8 and n = 9. Then we use an inductive argument to show that, independent of who starts, Maker can win the game if and only if n ≥ 8. This, in particular, resolves in the affirmative the long-standing conjecture of Papaioannou from [11].We also study two standard positional games related to Hamiltonicity game. For Hamiltonian Path game, we show that Maker can claim a Hamiltonian path if and only if n ≥ 5, independent of who starts. Next, we look at Fixed Hamiltonian Path game, where the goal of Maker is to claim a Hamiltonian path between two predetermined vertices. We prove that if Maker starts the game, he wins if and only if n ≥ 7, and if Breaker starts, Maker wins if and only if n ≥ 8. Using this result, we are able to improve the previously best upper bound on the smallest number of edges a graph on n vertices can have, knowing that Maker can win the Maker-Breaker Hamiltonicity game played on its edges.To resolve the outcomes of the mentioned games on small (finite) boards, we devise algorithms for efficiently searching game trees and then obtain our results with the help of a computer.

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Building spanning trees quickly in Maker-Breaker games

Cited by 8 publications

References 19 publications

Generating random graphs in biased Maker-Breaker games

Generating random graphs in biased Maker-Breaker games

Fast Strategies in Waiter-Client Games

Hamiltonian Maker–Breaker Games on Small Graphs

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