Online sum-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 uncolored vertices, scoring ∣M ∣ points. Painter then gives a color to a subset of M that is independent in G. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of G, written s̊(G). The game is an online variant of online sum list coloring. We prove ∣V (G)∣ 2α(G) + 12 ≤ s̊(G) ∣V (G)∣ ≤max{ ∣V (H)∣ α(H) ∶H ⊂ G}, where α(G) is the independence number, and we study when equality holds in the bounds. We compute s̊(G) for graphs with α(G) = 2. Among n-vertex trees, we prove that s̊ is minimized by the star and maximized by the path. We also obtain good bounds on s̊(Kr,s).