The Brush Number of the Two-Dimensional Torus
Ta Sheng Tan
TL;DR
This paper determines the brush number of the two-dimensional torus and the cartesian product of a clique with a path, addressing open questions in graph cleaning processes and related conjectures.
Contribution
It provides the first exact values for the brush number of the two-dimensional torus and extends understanding of graph cleaning in product graphs.
Findings
Brush number of the 2D torus is explicitly calculated.
Brush number of clique-path product is established.
Results answer open questions in graph cleaning theory.
Abstract
In this paper we are interested in the brush number of a graph - a concept introduced by McKeil and by Messinger, Nowakowski and Pralat. Our main aim in this paper is to determine the brush number of the two-dimensional torus. This answers a question of Bonato and Messinger. We also find the brush number of the cartesian product of a clique with a path, which is related to the Box Cleaning Conjecture of Bonato and Messinger.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Graph Theory Research · Geometric and Algebraic Topology