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

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 $n\ge 2$ vertices namely: $b_r(\mu (G))=b_r({\mu }^{\to }(G))=2\sum _{i=1}^n{d}_{G_{b_r(G)}^{\to }}^{+}(v_i).$

5 pages. arXiv admin note: substantial text overlap with arXiv:1501.01381