The Game Chromatic Index of Forests of Maximum Degree 5

Abstract

Using a refinement of the methods of Erdös et al. [6] we prove that the game chromatic index of forests of maximum node degree 5 is at most 6. This improves the previously known upper bound 7 for this parameter. The bound 6 is tight [6].

The Relaxed Edge-Coloring Game and k-Degenerate Graphs

“…It was shown that if ∆ = 3 or ∆ ≥ 6, then any tree T with ∆(T ) = ∆ satisfies χ ′ g (T ) ≤ ∆+1. In [1], this result was extended to trees (and forests) with ∆ = 5. The case for ∆ = 4 remains open, indicating the difficulty of determining these parameters, even in the non-relaxed environment.…”

The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices

“…We consider in this paper an edge-coloring game studied in [1,2,6,7,8,9]. In the game, two players, Alice and Bob, alternately select a color from a set of colors and put it on an uncolored edge of an initially uncolored, finite and simple graph G such that adjacent edges receive distinct colors.…”

“…Erdös et al [7] showed that best possible upper bound for the class of trees T with maximum degree ∆(T ) is at least ∆(T ) + 1 if ∆(T ) ≥ 2. The upper bound ∆(T ) + 1 holds when ∆(T ) = 3 [1,5] or ∆(T ) ≥ 5 [2,7].…”

“…The game studied in this paper is under the game variant that Bob begins and has the right to skip any number of turns. This variant was proposed by Andres [2]. Andres [2] proposed six game variants in which either Alice or Bob could take the first move; and either Alice, Bob, or none of them could be allowed to skip.…”

The incidence game chromatic number

“…Upper bounds for the game chromatic number have been determined for forests [6], outerplanar graphs [7], graphs embeddable in an orientable surface [11], line graphs of k-degenerate graphs [4], line graphs of forests with maximum degree ∆ = 4 [4,5,1], and planar graphs [11,15].…”