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arXiv · 1612.04702

Online Sum-Paintability: Slow-Coloring of Trees

Gregory J. Puleo·Douglas B. West
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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.

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Gregory J. Puleo, Douglas B. West. 2016-12-14. Online Sum-Paintability: Slow-Coloring of Trees. https://arxiv.org/abs/1612.04702

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