The edge coloring game on trees with the number of colors greater than the game chromatic index

Let X∈{A,B}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\in \{A,B\}$$\end{document} and Y∈{A,B,-}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y\in \{A,B,-\}$$\end{document}, where A, B and − denote (player) Alice, (player) Bob and none of the players, respectively. In the k-[X, Y]-edge-coloring game, Alice and Bob alternately choose a color from a given color set with k colors to color an uncolored edge of a graph G such that no adjacent edges receive the same color. Player X begins and Player Y has the right to skip any number of turns. Alice wins the game if all the edges of G are finally colored; otherwise, Bob wins. The [X, Y]-game chromatic index of an uncolored graph G, denoted by χ[X,Y]′(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi '_{[X,Y]}(G)$$\end{document}, is the least k such that Alice has a winning strategy for the game. We prove that, for any [X, Y], Alice has a winning strategy for the k-[X, Y]-edge-coloring game on any tree T when k>χ[X,Y]′(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>\chi '_{[X,Y]}(T)$$\end{document}. Moreover, using some parts of the proofs of the above results, we show that there is a tree T satisfying χ[A,-]′(T)<χ[B,-]′(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi '_{[A,-]}(T)<\chi '_{[B,-]}(T)$$\end{document} and χ[A,-]′(T-e)<χ[B,-]′(T-e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi '_{[A,-]}(T-e)<\chi '_{[B,-]}(T-e)$$\end{document} for some edge e of T. This solves an open problem proposed by Andres et al. (Discrete Appl Math 159:1660–1665, 2011).