[Paper Review] Embedding bounded degree spanning trees in random graphs

Richard Montgomery|arXiv (Cornell University)|May 26, 2014

Advanced Graph Theory Research14 references36 citations

TL;DR

This paper proves that a random graph $\mathcal{G}(n, \Delta \log^5 n / n)$ almost surely contains any spanning tree of order $n$ with maximum degree at most $\Delta$, using a novel absorption-based embedding method that combines sparse random graphs and reservoir-based path covering. The result improves prior bounds for bounded-degree spanning trees and provides a foundation for proving the full conjecture on tree universality in random graphs.

ABSTRACT

We prove that if a tree $T$ has $n$ vertices and maximum degree at most $Δ$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,Δ\log^5 n/n)$.

Motivation & Objective

• To close the gap between known embedding thresholds and Kahn's conjecture on the existence of all bounded-degree spanning trees in random graphs.
• To develop a robust embedding method that works for trees with long bare paths and those with many leaves, using absorption and reservoir techniques.
• To establish a threshold probability $p = \Delta \log^5 n / n$ that guarantees almost sure containment of any such spanning tree in $\mathcal{G}(n,p)$.
• To lay the groundwork for proving the full conjecture that $\mathcal{G}(n, C\log n / n)$ contains all $n$-vertex trees of maximum degree $\Delta$.
• To introduce a new reservoir construction using systems of paths, improving over single-path reservoirs in efficiency and applicability.

Proposed method

• Uses a two-pronged embedding strategy: one for trees with many leaves (via subtree embedding and matching), and another for trees with long bare paths (via path covering and absorption).
• Applies a directed graph version of the absorption method, constructing reversible paths as absorbers to enable flexible path extension.
• Employs a reservoir system composed of multiple disjoint paths rather than a single long path, enhancing robustness and reducing required edge density.
• Leverages expansion properties in sparse random graphs to construct paths between specified vertices using a modified version of Friedman and Pippenger's theorem on oriented tree embedding.
• Introduces a directed absorber structure based on reversible paths, allowing for both full and partial vertex inclusion in path systems.
• Relies on a general absorption framework inspired by Rödl, Ruciński, and Szemerédi, adapted to work in sparse random graphs with controlled expansion.

Experimental results

Research questions

• RQ1 Can the threshold probability for embedding all $n$-vertex trees of maximum degree $\Delta$ in $\mathcal{G}(n,p)$ be reduced below $\Delta \log^5 n / n$?
• RQ2 Can a universal embedding result be achieved for all bounded-degree spanning trees in $\mathcal{G}(n,p)$ with $p = \Delta \log^5 n / n$?
• RQ3 Can the absorption method be adapted to work effectively in sparse random graphs with controlled expansion properties?
• RQ4 Is it possible to cover the remaining vertices after embedding a core subtree using disjoint paths of fixed length, even at low edge density?
• RQ5 Can directed absorbers based on reversible paths improve the efficiency and reduce the required probability in tree embedding?

Key findings

• The random graph $\mathcal{G}(n, \Delta \log^5 n / n)$ almost surely contains every $n$-vertex tree with maximum degree at most $\Delta$.
• The proof establishes a universal embedding result for trees with long bare paths by revealing the entire graph in advance and using absorbers constructed in a sparse random graph.
• For trees with many leaves, the method reveals additional edges after embedding a core subtree, enabling a matching-based completion of the embedding.
• The reservoir is constructed as a system of disjoint paths, which allows for more efficient and flexible path covering than single-path reservoirs.
• The method achieves a natural barrier at $p \approx \Delta \log^2 n / n$, suggesting further technical developments are needed to reach the conjectured threshold of $C\log n / n$.
• The paper provides a foundation for a future proof of Kahn's conjecture, with a forthcoming work expected to show universality at $p = f(\Delta)\log^2 n / n$.