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

Let $$X\in \{A,B\}$$ X ∈ { A , B } and $$Y\in \{A,B,-\}$$ Y ∈ { A , B , - } , 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 $$\chi '_{[X,Y]}(G)$$ χ [ X , Y ] ′ ( G ) , 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 $$k>\chi '_{[X,Y]}(T)$$ k > χ [ X , Y ] ′ ( T ) . Moreover, using some parts of the proofs of the above results, we show that there is a tree T satisfying $$\chi '_{[A,-]}(T)