A biased edge coloring game
Runze Wang
TL;DR
This work introduces the $(m,1)$-edge coloring game, a biased Maker–Breaker variant of the edge coloring game, and defines the game chromatic index $\chi'_g(G;m,1)$ as the minimum number of colors for Maker to win. It develops general techniques and bounds, proving $\chi'_g(T;m,1)\le\Delta(T)+2$ for trees and improving to $\le\Delta(T)+1$ when $m\ge diam(T)-2$, with constructive strategies based on directed rooted trees. For caterpillars and wheels, the authors determine exact indices: caterpillars with $\Delta(T)\ge4$ satisfy $\chi'_g(T;m,1)=\Delta(T)$, while wheels satisfy $\chi'_g(W_n;m,1)=n$ for $n\ge5$ (with detailed small-$n$ cases and a notable nonmonotonicity $\chi'_g(W_4;3,1)=5>4=\chi'_g(W_4;2,1)$). The paper also discusses monotonicity phenomena, showing that larger $m$ does not always reduce the index and proposing a conjecture about when monotonicity fails, thereby deepening understanding of how move budgets interact with graph structure in edge coloring games.
Abstract
We combine the ideas of edge coloring games and asymmetric graph coloring games and define the \emph{$(m,1)$-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 \emph{$(m,1)$-game chromatic index} of $G$ to be the smallest nonnegative integer $k$ such that Maker has a winning strategy with $|X|=k$. 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.