Online Paintability: The Slow-Coloring Game

The slow-coloring game is played by Lister and Painter on a graph G. On each round, Lister marks a nonempty subsetM of the remaining vertices, scoring |M | points. Painter then deletes a subset of M that is independent in G. The game ends when all vertices are deleted. Painter’s goal is to minimize the total score; Lister seeks to maximize it. The score that each player can guarantee doing no worse than is the sum-color cost of G, written s̊(G). The game is a variant of online list coloring. We prove several results. Always |V (G)| 2α(G) + 1 2 ≤ s̊(G) |V (G)| ≤ max { |V (H)| α(H) : H ⊂ G } , where α(G) is the independence number. Among n-vertex trees, s̊ is minimized by the star and maximized by the path. Trivially s̊(G) is at most the sum-paintability of G, with equality only when all components of G are complete. Also s̊(G) is at least the chromatic sum (the minimum sum of vertex colors in a proper coloring by positive integers), with equality if α(G) ≤ 2. Finally, we give good bounds on s̊(Kr,s).