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

The game chromatic number of trees and forests

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4.Comment: 18 pages, 15 figures, submitte

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

Edge coloring of a graph

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2004Includes bibliographical references (leaves: 35-36)Text in English; Abstract: Turkish and Englishviii, 36 leavesThe edge coloring problem is one of the fundamental problem on graphs which often appears in various scheduling problems like the le transfer problem on computer networks. In this thesis, we survey old and new results on the classical edge coloring as well as the generalized edge coloring problems. In addition, we developed some algorithms and modules by using Combinatorica package to color the edges of graphs with webMathematica which is the new web-based technology

Digraph Coloring Games and Game-Perfectness

In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix

Martin Gardner and His Influence on Recreational Math

Recreational mathematics is a relatively new field in the world of mathematics. While sometimes overlooked as frivolous since those who practice it need no advanced knowledge of the subject, recreational mathematics is a perfect transition for people to experience the joy in logically establishing a solution. Martin Gardner recognized that this pattern of proving solutions to questions is how mathematics progresses. From his childhood on, Gardner greatly influenced the mathematical world. Although not a mathematician, he inspired many to pursue careers and make advancements in mathematics during his 25-year career with Scientific American. He encouraged novices to expand their knowledge, enlightened professionals of computer science developments, and established his own proofs