Homometric sets in trees

TL;DR

This paper investigates the existence and size of disjoint homometric vertex subsets in trees, providing improved bounds over previous results and specific bounds for haircomb trees.

Contribution

It proves that any tree contains disjoint homometric sets of size at least rac{\sqrt{n/2} - 1}{}, improving previous bounds and establishing larger sets in haircomb trees.

Findings

Existence of disjoint homometric sets of size rac{\sqrt{n/2} - 1}{} in any tree.

In haircomb trees, disjoint homometric sets of size rac{cn^{2/3}}{} for some constant c.

Improved bounds over previous results for the size of homometric sets in trees.

Abstract

Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least $cn^{2/3}$ for a constant $c > 0$.