Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, March 12, 2015, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Ralph Keusch

Colorability saturation games

Let P be any monotone increasing graph property. A graph G is P-saturated if G does not satisfy P, but together with any additional edge, G will have P. The saturation game (n,P) is the following game: Two players Mini and Maxi start with the empty graph on n vertices and take turns adding edges to the graph, such that P never holds. At some point, the graph is P-saturated and the game stops. Mini wants that the game is as short as possible, Maxi wants the opposite. The score s(n,P) denotes the number of edges at the end of the game, assuming both play optimally.
Results on the score of saturation games are quite scarce. We will discuss several examples and then focus on colorability saturation games, where P is "G has chromatic number at least k+1". For k=2, it is not difficult to see that the score is n^2/4. However, for larger values of k, the game becomes more complicated. In 2014, Hefetz, Krivelevich, Naor and Stojakovic proved (among bounds for other games) that for k=3, the score is at most 21n^2/64. We will show a matching lower bound. Hence, the score of this game is in fact 21n^2/64 + O(n).

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