Set-Sequential Labelings of Odd Trees

TL;DR

This paper advances the understanding of set-sequential labelings in odd trees, providing new constructions, partial results towards the Odd Tree Conjecture, and clarifications of related combinatorial theorems.

Contribution

It demonstrates that certain caterpillars are set-sequential, introduces methods to construct larger set-sequential graphs, and clarifies existing theorems related to binary vector partitions.

Findings

Certain caterpillars are proven to be set-sequential.

New constructions of set-sequential graphs from bipartite graphs are provided.

Clarifications are made on a theorem about partitioning binary vectors in 011.

Abstract

A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of the labels of the endpoints. It has been conjectured that all trees on $2^n$ vertices with only odd degree are set-sequential (the "Odd Tree Conjecture"), and in this paper, we present progress toward that conjecture. We show that certain kinds of caterpillars (with restrictions on the degrees of the vertices, but no restrictions on the diameter) are set-sequential. Additionally, we introduce some constructions of new set-sequential graphs from smaller set-sequential bipartite graphs (not necessarily odd trees). We also make a conjecture about pairings of the elements of $\mathbb{F}_2^n$ in a particular way; in the process, we provide a substantial clarification of a proof of a theorem that partitions $\mathbb{F}_2^n$ from a 2011 paper by Balister et al. Finally, we put forward a result on bipartite graphs that is a modification of a theorem in Balister et al.