Embedding Nearly Spanning Trees

TL;DR

This paper proves a near-spanning version of the Erdős-Sós Conjecture, showing that graphs with slightly more vertices than a tree of size k contain that tree as a subgraph, for large enough trees.

Contribution

It establishes a near-spanning result for the Erdős-Sós Conjecture, extending its validity to graphs slightly larger than the tree, for sufficiently large trees.

Findings

Validates the conjecture for large trees in nearly spanning graphs.

Identifies conditions under which the conjecture holds.

Provides bounds on graph size relative to the tree.

Abstract

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every tree $T$ with $k \ge k_0$ edges and every graph $G$ with $|V(G)| \le (1+\delta)|V(T)|$.