# On the Erdős-Sos conjecture

#### Abstract

We investigate a tantalizing problem in extremal graph theory known as the Erdős-Sós conjecture. The Erdős-Sós conjecture states that every simple graph with average degree greater than * k* − 2 contains every tree on *k* vertices as a subgraph (*k* is a positive integer). We prove various special cases each of which places certain restrictions on the class of graphs or the class of trees considered. In our descriptions below, *G* is a simple graph (that is, no loops and no multi-edges) on *n* vertices that has average degree greater than *k* − 2; and that *T* is any tree on *k* vertices. ^ In 1989, Sidorenko proved the conjecture holds if *T* has a vertex *v* with at least k2 − 1 leaf neighbors. In the first manuscript we improve upon this result by proving it is sufficient to assume that *T* has a vertex *v* with at least k2 − 2 leaf neighbors. We use this to prove the conjecture holds if the graph has minimum degree *k* − 4. From this result, we obtain that the conjecture holds for all *k* ≤ 8. ^ A spider of degree *d* is a tree that can be thought of as the union of *d* edge-disjoint paths that share exactly one common end-vertex. In the second manuscript, we prove that *G* contains every spider of degree *d* where *d* = 3 or *d* > 3k4 − 2. ^ In the third manuscript, we prove the conjecture holds if *G* has at most *k* + 3 vertices. ^ In the fourth manuscript, we prove the conjecture holds if *G * is P_{k}_{+4}-free, that is, if * G* contains no path on *k* + 4 vertices. ^

#### Subject Area

Mathematics

#### Recommended Citation

Gary F Tiner,
"On the Erdős-Sos conjecture"
(2007).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3277009.

http://digitalcommons.uri.edu/dissertations/AAI3277009