Game Chromatic Index of Graphs with Given Restrictions on Degrees

Given a graph $G$ and an integer $k$, two players alternatively color the edges of $G$ using $k$ colors so that adjacent edges get different colors. The game chromatic index $\chi_g’(G)$ is the minimum $k$ for which the first player has a strategy that ensures that all edges of $G$ get colored. The trivial bounds on the game chromatic index are $\Delta(G) \leq \chi’_g(G) \leq 2 \Delta(G) -1$ where $\Delta(G)$ is the maximal degree of $G$. We explore families of graphs at the two extremes.