Spanning trees in graphs without large bipartite holes

TL;DR

This paper proves that large graphs with certain minimum degree and no large bipartite holes contain all spanning trees with bounded maximum degree, extending previous results in graph theory.

Contribution

It establishes a new condition under which all bounded-degree spanning trees are guaranteed to exist in large graphs.

Findings

Graphs with minimum degree proportional to n contain all bounded-degree spanning trees.

Absence of large bipartite holes is crucial for embedding spanning trees.

Results strengthen previous theorems by Böttcher et al.

Abstract

We show that for any $\varepsilon>0$ and $\Delta\in\mathbb{N}$, there exists $\alpha>0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $\delta(G)\geq\varepsilon n$ and $e(X, Y)>0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$. This strengthens a result of B\"ottcher et al.