We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph Kn, 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, resolve...

the aim of this research is consider the importance of decision maker level of analysis in the study of foreign policy. this capability created by the regard to the role of man in foreign policy and international relations theories. however, the main research question is whether the decision maker level of analysis will be the theoretical support for the analysis of foreign policy and what are ...

For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T . We prove that if T has bounded maximum degree, then Maker can win this game within n+ 1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n−1 moves and provide non-trivial ...

A Graph Construction Game is a Maker-Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph KN . Maker’s aim is to claim a copy of a given target graph G while Breaker’s aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N > d2n, then Maker can claim a copy of G in at most d2n rounds....