Hamiltonian games

Lu, Xiaoyun

doi:10.1016/0095-8956(92)90030-2

Cited by 9 publications

(7 citation statements)

References 8 publications

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“…In this game Maker's aim is to create as many pairwise edge-disjoint Hamilton cycles as possible. Lu proved [13] that Maker can always produce at least 1 16 n Hamilton cycles and conjectured that Maker should be able to make ( 1 4 − )n for any fixed > 0. This conjecture follows immediately from our Theorem 1.1 and Theorem 2 of [11].…”

Section: Consequencesmentioning

confidence: 99%

The Game of JumbleG

Frieze¹,

Krivelevich²,

Pikhurko³

et al. 2005

Combinator. Probab. Comp.

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For Béla Bollobás on his 60th birthday JumbleG is a Maker-Breaker game. Maker and Breaker take turns in choosing edges from the complete graph K n. Maker's aim is to choose what we call an-regular graph (that is, the minimum degree is at least (1 2 −)n and, for every pair of disjoint subsets S, T ⊂ V of cardinalities at least n, the number of edges e(S, T) between S and T satisfies e(S,T) |S| |T | − 1 2 .) In this paper we show that Maker can create an-regular graph, for 2(log n/n) 1/3. We also consider a similar game, JumbleG2, where Maker's aim is to create a graph with minimum degree at least 1 2 − n and maximum co-degree at most 1 4 + n, and show that Maker has a winning strategy for > 3(log n/n) 1/2. Thus, in both games Maker can create a pseudo-random graph of density 1 2. This guarantees Maker's win in several other positional games, also discussed here.

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Section: Consequencesmentioning

confidence: 99%

The Game of JumbleG

Frieze¹,

Krivelevich²,

Pikhurko³

et al. 2005

Combinator. Probab. Comp.

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“…Hence, in such a game the family F is the intersection of a monotone increasing family and a monotone decreasing family. Avoider/Enforcer games were studied in [3,14,21,22,23]. Similarly to Maker/Breaker games, one would like to define for each monotone increasing family F the Avoider/Enforcer threshold bias f F .…”

Section: Introductionmentioning

confidence: 99%

Planarity, Colorability, and Minor Games

Hefetz¹,

Krivelevich²,

Stojaković³

et al. 2008

SIAM J. Discrete Math.

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Let m and b be positive integers and let F be a hypergraph. In an (m, b) Maker-Breaker game F two players, called Maker and Breaker, take turns selecting previously unclaimed vertices of F. Maker selects m vertices per move and Breaker selects b vertices per move. The game ends when every vertex has been claimed by one of the players. Maker wins if he claims all the vertices of some hyperedge of F; otherwise Breaker wins. An (m, b) Avoider-Enforcer game F is played in a similar way. The only difference is in the determination of the winner: Avoider loses if he claims all the vertices of some hyperedge of F; otherwise Enforcer loses. In this paper we consider the Maker-Breaker and Avoider-Enforcer versions of the planarity game, the k-colorability game and the K t-minor game.

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“…Therefore, Theorem 1 also implies the following: Theorem 2 [6]. If a ( G ) 5 A(G)n, then G is Hamiltonian.…”

Section: If a ( G ) 5 8(g)n Then G Is Hamiltonian Where A ( G ) Is Thementioning

confidence: 74%

“…If a ( G ) 5 8(G)n, then G is Hamiltonian, where a ( G ) is the If we define A(G) = min{(eG(A,Z))/lAI 1x1; 0 # A C V}, then it is easy to see that A(G) 5 8(G). Therefore, Theorem 1 also implies the following: Theorem 2 [6]. If a ( G ) 5 A(G)n, then G is Hamiltonian.…”

mentioning

confidence: 81%