Discrete Mathematics and Combinatorics
Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted χ gd (G). It is known that there exist graphs such that χ g0 (G) = 3, but χ g1 (G) > 3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m ≤ χ g0 (G) < χ g1 (G).
This is an electronic version of an article published in Order, 2012, volume 29, issue 3, pages 507-512. Order is available online at: doi:10.1007/s11083-011-9217-1.
Complete multipartite graphs and the relaxed coloring game.
Order, 2012, volume 29, issue 3, pages 507-512
Dunn, Charles, "Complete Multipartite Graphs and the Relaxed Coloring Game" (2012). Faculty Publications. Accepted Version. Submission 5.