Spanning trees homeomorphic to a small tree

A classical result of Ore states that if a graph GG of order nn satisfies degGx+degGy≥n−1degGx+degGy≥n−1 for every pair of nonadjacent vertices xx and yy of GG, then GG contains a hamiltonian path. In this note, we interpret a hamiltonian path as a spanning tree which is a subdivision of K2K2 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 kk, if a connected graph GG satisfies degGx+degGy≥n−kdegGx+degGy≥n−k for every pair of nonadjacent vertices xx and yy of GG, then GG contains a spanning tree which is a subdivision of a tree of order at most k+2k+2. We also discuss the sharpness of the result.