Spanning trees homeomorphic to a small tree.

A classical result of Ore states that if a graph G of order n satisfies deg G x + deg G y ≥ 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 K 2 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 deg G x + deg G y ≥ 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. [ABSTRACT FROM AUTHOR]