A Note on Weighted Rooted Trees

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

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∈A and b∈B, a and b are unrelated. Let ω be a nonnegative weight function defined on V(T) with Σv∈V(T)ω(v)=1. In this note, we prove that either there is an (r,u)-path P with Σv∈V(P)ω(v)≥1/3 for some u∈V(T), or there exist unrelated sets A,B⊆V(T) such that Σa∈Aω(a)≥1/3 and Σb∈Bω(b)≥1/3. The bound 1/3 is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé

Oasis 2016

Literary Magazine with artwork, photographs submitted by students.English and Modern Languages DepartmentAngelo State Universit