We consider unbiased Maker-Breaker games played on the edge set of the complete graph K n 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 Erdős [V. Chvátal, P. Erdős, 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))kn/2 moves, and this is obviously asymptotically tight. Several other related results are derived as well. (D. Hefetz), krivelev@post.tau.ac.il (M. Krivelevich), smilos@inf.ethz.ch (M. Stojaković), szabo@inf.ethz.ch (T. Szabó).
An antimagic labeling of an undirected graph G with n vertices and m edges is a bijection from the set of edges of G to the integers {1, . . . , m} such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labeling. In (N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990, pp. 108-109), Hartsfield and Ringel conjectured that every simple connected graph, other than K 2 , is antimagic. Despite considerable effort in recent years, this conjecture is still open. In this article we study a natural variation; namely, we consider antimagic labelings of directed graphs. In particular, we prove that every directed graph whose underlying undirected graph is "dense" is antimagic, and that almost every undirected d-regular graph admits an orientation which is antimagic.
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of "small" sets, the other ensuring the existence of an edge between any two disjoint "large" sets. We also discuss applications in positional games, random graphs and extremal graph theory.
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