Spanning trees with large maximum degrees

TL;DR

The work addresses optimal minimum-degree conditions for embedding every spanning tree with large maximum degree into an $n$-vertex host. By extending Komlós–Sárközy–Szemerédi-type results, it shows that when $\Delta \gg n/\log n$, the asymptotically tight threshold is $\delta(G) \ge n-n^{1-(1+o(1))\Delta/n}$, with corresponding random-graph analogues. The proof blends two regime analyses (high-range and low-range) using auxiliary lemmas on trees with many bare paths and many leaves, Hall-type matchings, and a randomized embedding framework. These results close gaps between deterministic and probabilistic thresholds for large-degree spanning trees and provide precise asymptotics in the random graph setting.

Abstract

The celebrated result of Komlós, Sárközy, and Szemerédi states that for any $\varepsilon>0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $δ(G)\geq(1/2+\varepsilon)n$ contains every $n$-vertex tree with maximum degree at most $cn/\log n$. This is best possible up to the value of $c$. In this paper, we extend this result to trees with higher maximum degrees, and prove that for $Δ\gg n/\log n$, roughly speaking, $δ(G)\geq n-n^{1-(1+o(1))Δ/n}$ is the asymptotically optimal minimum degree condition which guarantees that $G$ contains every $n$-vertex spanning tree with maximum degree at most $Δ$. We also prove the corresponding statements in the random graph setting.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.2
• proof
• Theorem 1.3
• proof
• Corollary 1.4
• proof
• Lemma 2.1: Azuma's Inequality W
• Lemma 2.2: Chernoff Bound JLR
• Lemma 2.3: Chernoff Bound