On the Erdös-Sós conjecture and graphs with large minimum degree
Date of Original Version
Suppose G is a simple graph with average vertex degree greater than k - 2. Erdös and Sós conjectured that G contains every tree on k vertices. Sidorenko proved G contains every tree that has a vertex v with at least [k/2] -1 leaf neighbors. We prove this is true if v has only [k/2] - 2 leaf neighbors. We generalize Sidorenko's result by proving that if G has minimum degree d, then G contains every tree that has a vertex with least (k-1) - d leaf neighbors. We use these results to prove that if G has average degree greater than k - 2 and minimum degree at least k - 4, then G contains every tree on k vertices.
Eaton, Nancy, and Gary Tiner. "On the Erdös-Sós conjecture and graphs with large minimum degree." Ars Combinatoria 95, (2010): 373-382. https://digitalcommons.uri.edu/math_facpubs/115