Refinements of degree conditions for the existence of a spanning tree without small degree stems

A spanning tree of a graph without no vertices of degree 2 is called a homeomorphically irreducible spanning tree (or a HIST) of the graph. Albertson, Berman, Hutchinson and Thomassen [J. Graph Theory 14 (1990), 247–258] gave a minimum degree condition for the existence of a HIST, and recently, Ito and Tsuchiya [J. Graph Theory 99 (2022), 162–170] found a sharp degree-sum condition for the existence of a HIST. In this paper, we refine these results, and extend the first one to a spanning tree in which no vertex other than the endvertices has small degree.