Spanning trees homeomorphic to a small tree

Abstract

A classical result of Ore states that if a graph G of order n satisfies degGx+degGy≥n−1 for every pair of nonadjacent vertices x and y of G, then G contains a hamiltonian path. In this note, we interpret a hamiltonian path as a spanning tree which is a subdivision of K2 and extend Ore’s result to a sufficient condition for the existence of a spanning tree which is a subdivision of a tree of a bounded order. We prove that for a positive integer k, if a connected graph G satisfies degGx+degGy≥n−k for every pair of nonadjacent vertices x and y of G, then G contains a spanning tree which is a subdivision of a tree of order at most k+2. We also discuss the sharpness of the result.