A Note on Weighted Rooted Trees

TL;DR

This paper proves a tight bound related to weighted paths and unrelated sets in rooted trees, answering a recent open question and providing a fundamental insight into the structure of weighted trees.

Contribution

It establishes a new tight bound on the distribution of weights in rooted trees, linking paths and unrelated sets, and resolves a recent open problem.

Findings

Either a weighted path with at least one-third of total weight exists

Or two unrelated sets each with at least one-third of total weight exist

The bound of one-third is proven to be tight

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 $\sum_{v\in V(P)}\omega(v)\ge \frac13$ for some $u\in V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in A }\omega(a)\ge \frac13$ and $\sum_{b\in B }\omega(b)\ge \frac13$. The bound $\frac13$ is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomass\'e.