Degree sum conditions for the existence of homeomorphically irreducible spanning trees

In 1990, Albertson, Berman, Hutchinson, and Thomassen proved a theorem which gives a minimum degree condition for the existence of a spanning tree with no vertices of degree 2. Such a spanning tree is called a homeomorphically irreducible spanning tree (HIST). In this paper, we prove that every graph of order n ( n >= 8) contains a HIST if d ( u ) + d ( v ) >= n - 1 for any nonadjacent vertices u and v. The degree sum condition is best possible.