Edge colouring Game on Trees with maximum degree $\Delta=4$

Consider the following game. We are given a tree $T$ and two players (say) Alice and Bob who alternately colour an edge of a tree (using one of $k$ colours). If all edges of the tree get coloured, then Alice wins else Bob wins. Game chromatic index of trees of is the smallest index $k$ for which there is a winning strategy for Alice. If the maximum degree of a node in tree is $\Delta$, Erdos et.al.[6], show that the game chromatic index is at least $\Delta+1$. The bound is known to be tight for all values of $\Delta\neq 4$. In this paper we show that for $\Delta=4$, even if Bob is allowed to skip a move, Alice can always choose an edge to colour and win the game for $k=\Delta+1$. Thus the game chromatic index of trees of maximum degree $4$ is also $5$. Hence, game chromatic index of trees of maximum degree $\Delta$ is $\Delta+1$ for all $\Delta\geq 2$. Moreover,the tree can be preprocessed to allow Alice to pick the next edge to colour in $O(1)$ time. A result of independent interest is a linear time algorithm for on-line edge-deletion problem on trees.