A note on the Brush Numbers of Mycielski Graphs, $\mu(G)$

TL;DR

This paper investigates the brush number of Mycielski graphs, providing a formula that relates it to an optimal orientation of the original graph, thus advancing understanding of graph cleaning processes.

Contribution

It establishes a simple formula for the brush number of Mycielski graphs based on optimal orientations of the original graph.

Findings

Brush number of Mycielski graphs equals twice the sum of out-degrees in an optimal orientation.

Provides a general formula for the brush number in terms of graph orientations.

Simplifies calculation of brush numbers for Mycielski graphs.

Abstract

The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielskian graph $\mu(G)$ of a simple connected graph $G$ to find $b_r(\mu(G))$ in terms of an \emph{optimal orientation} of $G$. We prove a surprisingly simple general result for simple connected graphs on $n \geq 2$ vertices namely: $b_r(\mu(G))= b_r(\mu^{\rightarrow}(G)) = 2\sum\limits_{i=1}^{n}d^+_{G^{\rightarrow}_{b_r(G)}}(v_i).$