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Fong et al. (The game chromatic index of some trees with maximum degree fourand adjacent degree-four vertices, J. Comb Optim 36 (2018) 1-12) proved thatthe game chromatic index of any tree $T$ of maximum degree 4 whose degree-fourvertices induce a forest of paths of length $l$ less than 2 is at most 5. Inthis paper, we show that the bound 5 is also valid for $l\leq 2$. Thispartially solves the problem of characterization of the trees whose gamechromatic index exceeds the maximum degree by at most 1, which was proposed byCai and Zhu (Game chromatic index of $k$-degenerate graphs, J. Graph Theory 36(2001) 144-155).

The game chromatic index of trees of maximum degree 4 with at most three degree-four vertices in a row