New Classes of Set-Sequential Trees

TL;DR

This paper introduces new classes of set-sequential trees by resolving specific cases of a conjecture, demonstrating that many caterpillars with certain properties are set-sequential, and proposing a recursive construction method.

Contribution

It proves that all certain caterpillars are set-sequential and presents a novel recursive method for constructing set-sequential trees.

Findings

All caterpillars of diameter ≤ 18 are set-sequential.

Caterpillars with at least 2^{k-1} vertices are set-sequential.

A new recursive construction method for set-sequential trees.

Abstract

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the label for either a single vertex or a single edge. We resolve certain cases of a conjecture of Balister, Gyori, and Schelp in order to show many new classes of trees to be set-sequential. We show that all caterpillars $T$ of diameter $k$ such that $k \leq 18$ or $|V(T)| \geq 2^{k-1}$ are set-sequential, where $T$ has only odd-degree vertices and $|T| = 2^{n-1}$ for some positive integer $n$. We also present a new method of recursively constructing set-sequential trees.