Graham's Tree Reconstruction Conjecture and a Waring-Type Problem on Partitions

TL;DR

This paper investigates Graham's Tree Reconstruction Conjecture, showing that a super-polynomial number of trees can be distinguished by their associated integer sequences, using combinatorial and number-theoretic techniques.

Contribution

It establishes a lower bound on the number of trees distinguishable by their sequences and introduces a novel construction using partitions from the Prouhet-Tarry-Escott problem.

Findings

Number of distinguishable trees is at least exponential in (a0(log n)^{3/2})

Constructs large collections of caterpillar graphs using advanced partition techniques

Provides new insights into the complexity of the Tree Reconstruction Conjecture

Abstract

Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that $G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$, $|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of the graph $H$. Little is known about this question apart from a few simple observations. We show that the number of trees on $n$ vertices which can be distinguished by their associated integer sequences is $e^{\Omega((\log n)^{3/2})}$. The proof strategy involves constructing a large collection of caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott problem.