The relaxed game chromatic index of k-degenerate graphs

The relaxed game chromatic index of k-degenerate graphs

The (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)-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 (G) = , then the first player, Alice, has a winning strategy for this game with...

Game chromatic index of k-degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving ®rst, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when ...

The d-relaxed game chromatic index of k-degenerated graphs

Strong chromatic index of k-degenerate graphs

A strong edge coloring of a graph G is a proper edge coloring in which every color class is an induced matching. The strong chromatic index χs(G) of a graph G is the minimum number of colors in a strong edge coloring of G. In this note, we improve a result by Dębski et al. [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a k-degenerate gra...

The Relaxed Edge-Coloring Game and k-Degenerate Graphs

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 [10...