Tree tilings in random regular graphs

TL;DR

This paper proves that large random regular graphs almost surely contain factors of all small trees up to a certain size, and provides algorithms to find these factors efficiently.

Contribution

It establishes the existence of tree factors in random regular graphs for sizes up to a specific bound and introduces efficient randomized and deterministic algorithms to find them.

Findings

Random regular graphs contain T-factors for all small trees T up to a size limit.

The algorithms to find T-factors run in near-linear expected time.

The results are tight, matching known non-existence thresholds for larger trees.

Abstract

We show that for every $\epsilon>0$ 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 $(1-\epsilon)d/\ln d$ vertices. This is best possible since, for large enough integer $d$, whp $G(n,d)$ does not contain a $\frac{(1+\epsilon)d}{\ln d}$-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.