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 $\mathrm{\mid }M\mathrm{\mid }$ 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 $\mathring{\mathrm{s}}(G)$. We develop a linear-time algorithm to compute $\mathring{\mathrm{s}}(G)$ 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.