On the Erdős-Sos conjecture
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 Pk+4-free, that is, if G contains no path on k + 4 vertices. ^
Gary F Tiner,
"On the Erdős-Sos conjecture"
Dissertations and Master's Theses (Campus Access).