A new upper bound on the game chromatic index of graphs

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, 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 $\chi'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $\Delta(G) \le \chi_g'(G) \le 2\Delta(G)-1$ hold for every graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every $\delta>0$ there exists a constant $c>0$ such that $\chi'_g(G) \le (2-c)\Delta(G)$ holds for any graph with $\Delta(G) \ge (\frac{1}{2}+\delta)v(G)$, and conjectured that the same holds for every graph $G$. In this paper, we show that $\chi'_g(G) \le (2-c)\Delta(G)$ is true for all graphs $G$ with $\Delta(G) \ge C \log v(G)$. In addition, we consider 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.Comment: 17 page

The Game Chromatic Number of Complete Multipartite Graphs with No Singletons

In this paper we investigate the game chromatic number for complete multipartite graphs. We devise several strategies for Alice, and one strategy for Bob, and we prove their optimality in all complete multipartite graphs with no singletons. All the strategies presented are computable in linear time, and the values of the game chromatic number depend directly only on the number and the sizes of sets in the partition

On game chromatic number analogues of Mycielsians and Brooks' Theorem

The vertex coloring game is a two-player game on a graph with given color set in which the first player attempts to properly color the graph and the second attempts to prevent a proper coloring from being achieved. The smallest number of colors for which the first player can win no matter how the second player plays is called the game chromatic number of the graph. In this paper we initiate the study of game chromatic number for Mycielskians and a game chromatic number analogue of Brooks' Theorem (which characterizes graphs for which chromatic number is at most the maximum degree of the graph). In particular, we determine the game chromatic number of Mycielskians of complete graphs, complete bipartite graphs, and cycles. In the direction of Brooks' Theorem, we show that if there are few vertices of maximum degree or if all vertices of maximum degree are at least three edges apart, then the game chromatic number is at most the maximum degree of the grap