On Homeomorphically Irreducible Spanning Trees in Cubic Graphs

Abstract

A spanning tree without a vertex of degree two is called a Hist which is an abbreviation for homeomorphically irreducible spanning tree. We provide a necessary condition for the existence of a Hist in a cubic graph. As one consequence, we answer affirmatively an open question on Hists by Albertson, Berman, Hutchinson and Thomassen.