Discrete Mathematics and Combinatorics
The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in . More broadly, we generalize the main result in  by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j).
The final publication is available at Springer via http://dx.doi.org/10.1007/s11083-014-9336-6.
Charles Dunn, David Morawski, & Jennifer Firkins Nordstrom
The relaxed edge-coloring game and k-degenerate graphs
Order, 2015, volume 32, issue 3, pages 347-361
Dunn, Charles; Morawski, David; and Nordstrom, Jennifer Firkins, "The Relaxed Edge-Coloring Game and k-Degenerate Graphs" (2015). Faculty Publications. Accepted Version. Submission 8.