Note on the game chromatic index of trees

We study edge coloring games defining the so-called game chromatic index of a graph. It has been reported that the game chromatic index of trees with maximum degree $\Delta = 3$ is at most $\Delta + 1$. We show that the same holds true in case $\Delta \geq 6$, which would leave only the cases $\Delta = 4$ and $\Delta = 5$ open. \u

The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-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 r = ∆+k−1 and d≥2k2 + 4k

The Relaxed Edge-Coloring Game and \u3cem\u3ek\u3c/em\u3e-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]. More broadly, we generalize the main result in [10] 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)

Edge-coloring via fixable subgraphs

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure

A new upper bound on the game chromatic index of graphs

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $\chi'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $\Delta(G) \le \chi_g'(G) \le 2\Delta(G)-1$ hold for every graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every $\delta>0$ there exists a constant $c>0$ such that $\chi'_g(G) \le (2-c)\Delta(G)$ holds for any graph with $\Delta(G) \ge (\frac{1}{2}+\delta)v(G)$, and conjectured that the same holds for every graph $G$. In this paper, we show that $\chi'_g(G) \le (2-c)\Delta(G)$ is true for all graphs $G$ with $\Delta(G) \ge C \log v(G)$. In addition, we consider a biased version of the game where Breaker is allowed to color $b$ edges per turn and give bounds on the number of colors needed for Maker to win this biased game.Comment: 17 page

The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs