On Erdős-Sós Conjecture for Trees of Large Size

On Erdős-Sós Conjecture for Trees of Large Size

Erdős and Sós conjectured that every graph G of average degree greater than k−1 contains every tree of size k. Several results based upon the number of vertices in G have been proved including the special cases where G has exactly k+1 vertices (Zhou), k + 2 vertices (Slater, Teo and Yap), k + 3 vertices (Woźniak) and k + 4 vertices (Tiner). We further explore this direction. Given an arbitrary ...

The Erdős-Sós Conjecture for Geometric Graphs

Let f(n, k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that 1 2 n k 1 n 2  f(n, k)  2 n(n 2) k 2 . For the case when k = n, we show that 2  f(n, n)  3. For the case when k = n and G is a geometric graph on a set of points in con...

The Erdős-Sós conjecture for spiders of diameter 9

The Loebl-Komlós-Sós Conjecture for Trees of Diameter 5 and for Certain Caterpillars

Loebl, Komlós, and Sós conjectured that if at least half the vertices of a graph G have degree at least some k ∈ N, then every tree with at most k edges is a subgraph of G. We prove the conjecture for all trees of diameter at most 5 and for a class of caterpillars. Our result implies a bound on the Ramsey number r(T, T ) of trees T, T ′ from the above classes.

On the Loebl-Komlós-Sós conjecture

The Loebl-Komlós-Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k (k ≥ n− 3).