Spanning trees without adjacent vertices of degree 2

TL;DR

This paper proves that sufficiently connected graphs always have spanning trees avoiding adjacent degree-2 vertices, and that graphs with minimum degree 3 always have spanning trees without three consecutive degree-2 vertices, extending previous results.

Contribution

It establishes new bounds on minimum degree ensuring the existence of spanning trees without adjacent or three consecutive degree-2 vertices.

Findings

Existence of a minimum degree d for spanning trees without adjacent degree-2 vertices

Graphs with minimum degree at least 3 have spanning trees without three consecutive degree-2 vertices

Extension of previous results on spanning trees and degree constraints

Abstract

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2.