Online Sum-Paintability: Slow-Coloring of Trees

TL;DR

This paper introduces a linear-time algorithm for computing the sum-color cost and interactive sum choice number of trees in the slow-coloring game, providing a complete characterization of extremal trees.

Contribution

It develops an efficient algorithm for calculating sum-color cost on trees and characterizes extremal trees, also computing the interactive sum choice number for trees.

Findings

Linear-time algorithm for sum-color cost on trees

Characterization of trees with extremal sum-color costs

Computation of interactive sum choice number for trees

Abstract

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