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

## TL;DR

This paper extends previous results on the game chromatic index of trees with maximum degree 4, showing that the bound of 5 colors applies even when degree-four vertices form paths of length up to 2.

## Contribution

It proves that the bound of 5 colors holds for trees with degree-four vertices forming paths of length up to 2, advancing the characterization of trees with small differences between chromatic index and maximum degree.

## Key findings

- The bound of 5 colors applies for trees with degree-four vertices in paths of length up to 2.
- This result partially addresses a problem on characterizing trees with game chromatic index close to maximum degree.

## Abstract

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