The relaxed game chromatic index of k-degenerate graphs

The (r,d)(r,d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G   called the (r,d)(r,d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k  -degenerate with Δ(G)=ΔΔ(G)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ+k-1r=Δ+k-1 and d⩾2k2+4kd⩾2k2+4k.