A new strategy for finding spanning trees without small degree stems

TL;DR

This paper introduces a new method for finding spanning trees with specific degree constraints, extending existing conditions for their existence and exploring degree-product conditions.

Contribution

It presents a novel strategy for identifying $[2,k]$-STs and refines known degree-sum conditions for HISTs, also examining degree-product conditions.

Findings

Refined degree-sum conditions for HIST existence

Extended conditions for $[2,k]$-STs

Investigated degree-product conditions

Abstract

For an integer $k\geq 2$, a spanning tree of a graph without vertices of degree from $2$ to $k$ is called a {\it $[2,k]$-ST} of the graph. The concept of $[2,k]$-STs is a natural extension of a homeomorphically irreducible spanning tree (or HIST), which is a well-studied graph structure. In this paper, we give a new strategy for finding $[2,k]$-STs. By using the strategy, we refine or extend a known degree-sum condition for the existence of a HIST. Furthermore, we also investigate a degree-product condition for the existence of a $[2,k]$-ST.