A Note on Weighted Rooted Trees

Abstract

Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are unrelated. Let $\omega$ be a nonnegative weight function defined on $V(T)$ with ${\sum }_{v\in V(T)}\omega (v)=1$. In this note, we prove that either there is an $(r,u)$-path $P$ with ${\sum }_{v\in V(P)}\omega (v)\ge \frac{1}{3}$ for some $u\in V(T)$, or there exist unrelated sets $A,B\subseteq V(T)$ such that ${\sum }_{a\in A}\omega (a)\ge \frac{1}{3}$ and ${\sum }_{b\in B}\omega (b)\ge \frac{1}{3}$. The bound $\frac{1}{3}$ is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé.