A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices

TL;DR

This paper establishes a new neighborhood union condition ensuring the existence of a homeomorphically irreducible spanning tree (HIST) in large connected graphs, improving previous degree-sum conditions by Ito and Tsuchiya.

Contribution

It introduces a neighborhood union condition that guarantees a HIST in large graphs, refining earlier degree-sum criteria and identifying exceptional graph families.

Findings

HIST exists under the neighborhood union condition for graphs with n ≥ 270.

The result improves previous degree-sum conditions for HIST existence.

Certain exceptional graph families and cut-vertices of degree 2 are excluded.

Abstract

For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree $2$. In this paper, we show that if $G$ is a graph of order $n\ge 270$ and $|N(u)\cup N(v)|\geq\frac{n-1}{2}$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$, then $G$ has a HIST, unless $G$ belongs to three exceptional families of graphs or $G$ has a cut-vertex of degree $2$. This result improves the latest conclusion, due to Ito and Tsuchiya, that a HIST in $G$ can be guaranteed if $d(u)+d(v)\geq n-1$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$.