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arXiv · 2412.07128

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

Yibo Li·Fengming Dong·Xiaolan Hu·Huiqing Liu
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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 $G$.

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Yibo Li, Fengming Dong, Xiaolan Hu, Huiqing Liu. 2024-12-10. A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices. https://arxiv.org/abs/2412.07128

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