On the Erdős-Sós conjecture for graphs having no path with k+4k+4 vertices

When GG is a graph with average degree greater than k−2k−2, Erdős and Gallai proved that GG contains a path on kk vertices. Erdős and Sós conjectured that under the same condition, GG should contain every tree on kk vertices. Several results based upon the number of vertices in GG have been proved including the special cases where GG has exactly kk vertices (Zhou), k+1k+1 vertices (Slater, Teo and Yap), k+2k+2 vertices (Woźniak) and k+3k+3 vertices (Tiner). To strengthen these results, we will prove that the Erdős-Sós conjecture holds when the graph GG contains no path with k+4k+4 vertices (no restriction is imposed on the number of vertices of GG).