A variation of a conjecture due to Erdös and Sós

Erdös and Sós conjectured in 1963 that every graph G on n vertices with edge number e(G) > ½ (k . 1)n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n . 9/2 k 2 + 37/2 k+14 and every graph G on n vertices with e(G) > ½ (k . 1)n, we prove that there exists a graph G. on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.