A biased edge coloring game

Abstract

We combine the ideas of edge coloring games and asymmetric graph coloring games and define the $(m,1)$-edge coloring game, which is alternatively played by two players Maker and Breaker on a finite simple graph $G$ with a set of colors $X$. Maker plays first and colors $m$ uncolored edges on each turn. Breaker colors only one uncolored edge on each turn. They make sure that adjacent edges get distinct colors. Maker wins if eventually every edge is colored; Breaker wins if at some point, the player who is playing cannot color any edge. We define the $(m,1)$-game chromatic index of $G$ to be the smallest nonnegative integer $k$ such that Maker has a winning strategy with $\left|X\right|=k$. We give some general upper bounds on the $(m,1)$-game chromatic indices of trees, determine the exact $(m,1)$-game chromatic indices of some caterpillars and all wheels, and show that larger $m$ does not necessarily give us smaller $(m,1)$-game chromatic index.