Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, October 03, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Ralph Keusch

Edge-coloring graphs against an adversary

We consider the following two-player game: Maker and Breaker alternatively color the edges of a given graph G with k colors such that the partial coloring is proper. Maker's goal is to play such that at the end, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index χ'_g(G) is defined as the smallest integer k for which Maker has a winning strategy. Clearly, for every graph the game chromatic index lies between Δ(G) and 2Δ(G)-1, where Δ(G) denotes the maximum degree of G. However, despite looking innocent, this game is believed to be hard to analyze. Except for very special or sparse graphs such as forests, no substantial strategies are known.
In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that there exists c > 0 such that for every graph with Δ(G) > n/2 we have χ'_g(G) < (2-c)Δ(G), and conjectured that the same holds for every graph G. In this talk, we show that for every graph with maximum degree Δ(G) > C log(n) the conjecture is true. We do this by using a randomized strategy for Maker. In addition, we study a biased version of the game where Breaker is allowed to color b edges per turn, and give bounds on the number of colors needed for Maker to win this biased game.

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