Tree tilings in random regular graphs

TL;DR

This work establishes that in the random d-regular graph G(n,d) with large d, whp every tree T of size at most (1-ε)d/ln d admits a T-factor, and this threshold is tight up to stars. The authors develop a two-stage constructive strategy: first, they partition the vertex set into k = |V(T)| parts so that degrees across part-pairs follow a near-mean concentration, using an algorithmic Lovász Local Lemma to obtain a 'nice' partition; second, they prove Hall-type conditions ensuring a perfect matching between each adjacent pair of parts, which together yield a T-factor. The method yields a randomized algorithm with expected running time $O(n^{1+o(1)})$ and a deterministic variant, highlighting a near-linear-time route to spanning tree-factors in random regular graphs. The results substantially extend our understanding of tree-factors in the sparse regime and leverage a combination of LLL-based partitioning, Gao–McKay-type edge-distribution bounds, and Hall’s theorem to achieve a robust, constructive existence proof with practical algorithmic consequences.

Abstract

We show that for every $ε>0$ there exists a sufficiently large $d_0\in \mathbb{N}$ such that for every $d\ge d_0$, whp the random $d$-regular graph $G(n,d)$ contains a $T$-factor for every tree $T$ on at most $(1-ε)d/\ln d$ vertices. This is best possible since, for large enough integer $d$, whp $G(n,d)$ does not contain a $\frac{(1+ε)d}{\ln d}$-star-factor. Our method gives a randomised algorithm which whp finds said $T$-factor and whose expected running time is $O(n^{1+o(1)})$, as well as an efficient deterministic counterpart.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3: Theorem 1.2 of MT10, rephrased
• Corollary 3.3
• proof
• Theorem 4: Corollary 8 in gao2023subgraph