tailieunhanh - Báo cáo toán học: "On two problems regarding the Hamiltonian cycle game"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On two problems regarding the Hamiltonian cycle game. | On two problems regarding the Hamiltonian cycle game Dan Hefetz Sebastian Stich Institute of Theoretical Computer Science ETH Zurich CH-8092 Switzerland Department of Mathematics ETH Zurich CH-8092 Switzerland sstich@ Submitted Oct 24 2008 Accepted Feb 18 2009 Published Feb 27 2009 Mathematics Subject Classification 91A46 05C45 Abstract We consider the fair Hamiltonian cycle Maker-Breaker game played on the edge set of the complete graph Kn on n vertices. It is known that Maker wins this game if n is sufficiently large. We are interested in the minimum number of moves needed for Maker in order to win the Hamiltonian cycle game and in the smallest n for which Maker has a winning strategy for this game. We prove the following results 1 If n is sufficiently large then Maker can win the Hamiltonian cycle game within n 1 moves. This bound is best possible and it settles a question of Hefetz Krivelevich Stojakovic and Szabo 2 If n 29 then Maker can win the Hamiltonian cycle game. This improves the previously best bound of 600 due to Papaioannou. 1 Introduction Let F be a hypergraph. In the fair Maker-Breaker game F two players called Maker and Breaker take turns in claiming previously unclaimed vertices of F with Breaker going first. Each player claims one vertex per turn. Maker wins the game as soon as he claims all the vertices of some hyperedge of F. If by the time every vertex of F is claimed by some player Maker was not able to fully claim any hyperedge of F then Breaker wins the game. The game which differs from F only in the fact that Maker is the first player instead of Breaker will be denoted by FM. Let n be a positive integer and let Hn be the hypergraph whose vertices are the edges of Kn and whose hyperedges are the edge sets of all Hamiltonian cycles of Kn. In this paper we study the fair Maker-Breaker Hamiltonian cycle game Hn. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R28 1 Following 2 we define T Hn to be the .