TY - JOUR

T1 - Avoider-Enforcer games

AU - Hefetz, Dan

AU - Krivelevich, Michael

AU - Szabó, Tibor

N1 - Funding Information:
E-mail addresses: [email protected] (D. Hefetz), [email protected] (M. Krivelevich), [email protected] (T. Szabó). 1 This paper is a part of the author’s PhD under the supervision of Prof. Michael Krivelevich. 2 Research supported in part by USA-Israel BSF Grant 2002-133, and by grant 526/05 from the Israel Science Foundation.

PY - 2007/7

Y1 - 2007/7

N2 - Let p and q be positive integers and let H be any hypergraph. In a (p, q, H) Avoider-Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p, q, H) game. We then use this condition to show that Enforcer can win the (1, q) perfect matching game on K2 n for every q ≤ c n / log n for an appropriate constant c, and the (1, q) Hamilton cycle game on Kn for every q ≤ c n log log log log n / log n log log log n for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1, q) connectivity game on Kn. This result is quite surprising as it substantially differs from its Maker-Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285-291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.

AB - Let p and q be positive integers and let H be any hypergraph. In a (p, q, H) Avoider-Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p, q, H) game. We then use this condition to show that Enforcer can win the (1, q) perfect matching game on K2 n for every q ≤ c n / log n for an appropriate constant c, and the (1, q) Hamilton cycle game on Kn for every q ≤ c n log log log log n / log n log log log n for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1, q) connectivity game on Kn. This result is quite surprising as it substantially differs from its Maker-Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285-291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.

KW - Biased games

KW - Positional games

UR - http://www.scopus.com/inward/record.url?scp=34249705382&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2006.10.001

DO - 10.1016/j.jcta.2006.10.001

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AN - SCOPUS:34249705382

SN - 0097-3165

VL - 114

SP - 840

EP - 853

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 5

ER -