TY - JOUR
T1 - Fast winning strategies in Maker-Breaker games
AU - Hefetz, Dan
AU - Krivelevich, Michael
AU - Stojaković, Miloš
AU - Szabó, Tibor
N1 - Funding Information:
E-mail addresses: [email protected] (D. Hefetz), [email protected] (M. Krivelevich), [email protected] (M. Stojaković), [email protected] (T. Szabó). 1 This paper is a part of the author’s PhD thesis under the supervision of Professor Michael Krivelevich. 2 Research supported in part by USA-Israel BSF Grants 2002-133 and 2006-322, by Grant 526/05 from the Israel Science Foundation, and by a Pazy Memorial Award. 3 Partially supported by the Ministry of Science, Republic of Serbia, and Provincial Secretariat for Science, Province of Vojvodina.
PY - 2009/1
Y1 - 2009/1
N2 - We consider unbiased Maker-Breaker games played on the edge set of the complete graph Kn on n vertices. Quite a few such games were researched in the literature and are known to be Maker's win. Here we are interested in estimating the minimum number of moves needed for Maker in order to win these games. We prove the following results, for sufficiently large n: (1)Maker can construct a Hamilton cycle within at most n + 2 moves. This improves the classical bound of 2n due to Chvátal and Erdo{combining double acute accent}s [V. Chvátal, P. Erdo{combining double acute accent}s, Biased positional games, Ann. Discrete Math. 2 (1978) 221-228] and is almost tight;(2)Maker can construct a perfect matching (for even n) within n / 2 + 1 moves, and this is tight;(3)For a fixed k ≥ 3, Maker can construct a spanning k-connected graph within (1 + o (1)) k n / 2 moves, and this is obviously asymptotically tight. Several other related results are derived as well.
AB - We consider unbiased Maker-Breaker games played on the edge set of the complete graph Kn on n vertices. Quite a few such games were researched in the literature and are known to be Maker's win. Here we are interested in estimating the minimum number of moves needed for Maker in order to win these games. We prove the following results, for sufficiently large n: (1)Maker can construct a Hamilton cycle within at most n + 2 moves. This improves the classical bound of 2n due to Chvátal and Erdo{combining double acute accent}s [V. Chvátal, P. Erdo{combining double acute accent}s, Biased positional games, Ann. Discrete Math. 2 (1978) 221-228] and is almost tight;(2)Maker can construct a perfect matching (for even n) within n / 2 + 1 moves, and this is tight;(3)For a fixed k ≥ 3, Maker can construct a spanning k-connected graph within (1 + o (1)) k n / 2 moves, and this is obviously asymptotically tight. Several other related results are derived as well.
KW - Hamilton cycle
KW - Maker-Breaker games
KW - Perfect matching
KW - k-connected graph
UR - https://www.scopus.com/pages/publications/56549111956
U2 - 10.1016/j.jctb.2008.04.001
DO - 10.1016/j.jctb.2008.04.001
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AN - SCOPUS:56549111956
SN - 0095-8956
VL - 99
SP - 39
EP - 47
JO - Journal of Combinatorial Theory. Series B
JF - Journal of Combinatorial Theory. Series B
IS - 1
ER -