Online Sum-Paintability: Slow-Coloring of Trees

The slow-coloring game is played by Lister and Painter on a graph G. On each round, Lister marks a nonempty subset M of the remaining 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’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). We develop a linear-time algorithm to compute s̊(G) when G is a tree, enabling us to characterize the n-vertex trees with the largest and smallest values. Our algorithm also computes on trees the interactive sum choice number, a parameter recently introduced by Bonamy and Meeks.