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

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