On the Erdos-Sós conjecture for graphs having no path with k + 4 vertices
Date of Original Version
When G is a graph with average degree greater than k - 2, Erdos and Gallai proved that G contains a path on k vertices. Erdos 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 Erdos-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). © 2013 Elsevier B.V. All rights reserved.
Eaton, Nancy, and Gary Tiner. "On the Erdos-Sós conjecture for graphs having no path with k + 4 vertices." Discrete Mathematics 313, 16 (2013): 1621-1629. doi:10.1016/j.disc.2013.04.024.