Chvatal and Erdos proved in 1978 that in a two player game between Maker and Breaker, who take turns in selecting previously unclaimed edges of the complete graph on n vertices, Maker can create a Hamilton cycle, for all large enough n. Their groundbreaking paper, which launched the research in Maker-Breaker games, generated, explicitly or implicitly, a large variety of interesting questions about Hamiltonicity games, like:
-- How long does it take Maker to create a Hamilton cycle?
-- Who wins if Breaker claims b>1 edges in each turn?
-- What is the largest bias b for which Maker still wins?
-- What happens if the Hamiltonicity game is played on a board other than the edge set of K_n?
-- What is the sparsest board on which Maker still wins?
-- Who is the winner in the game played on the edge set of a random graph?
-- What about other variants of the Hamiltonicity game?
-- What can one say about the Avoider-Enforcer Hamiltonicity game (where Avoider wins by avoiding to create a Hamilton cycle of his edges by the end, and Enforcer wins otherwise)?
In this talk I will describe answers, partial of full, to the above questions, of which quite a few have been obtained relatively recently. The research in this fascinating subject can serve as an illustration of the development of the field of positional games over the years.