Packing Tree Factors in Random and Pseudo-Random Graphs

Deepak Bal,Alan Frieze,Michael Krivelevich,Po-Shen Loh

doi:10.37236/3285

Abstract

For a fixed graph $H$ with $t$ vertices, an $H$-factor of a graph $G$ with $n$ vertices, where $t$ divides $n$, is a collection of vertex disjoint (not necessarily induced) copies of $H$ in $G$ covering all vertices of $G$. We prove that for a fixed tree $T$ on $t$ vertices and $\epsilon>0$, the random graph $G_{n,p}$, with $n$ a multiple of $t$, with high probability contains a family of edge-disjoint $T$-factors covering all but an $\epsilon$-fraction of its edges, as long as $\epsilon^4 n p \gg \log^2 n$. Assuming stronger divisibility conditions, the edge probability can be taken down to $p>\frac{C\log n}{n}$. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

Highlights

Let H be a graph with t vertices and let n be divisible by t
> 0, the random graph Gn,p, with n a multiple of t, with high probability contains a family of edge-disjoint T -factors covering all but an -fraction of its edges, as long as 4np log2 n
H-factors have been an important object in the study of random graphs

Summary

Introduction

Let H be a graph with t vertices and let n be divisible by t. In intermediate papers such as [4] and [10], the notion of approximate or almost optimal packings was studied The results in these papers state that for certain ranges for p, all but a vanishing fraction of the edges of Gn,p can be covered with edge-disjoint Hamilton cycles. Note that if one applies it with the particular choice τ = τ0, the conclusion is that there is a real C0 such that Gn,p satisfies P( ) whp for τ0 | n, when p> This is within a factor (C0) of the best possible result, and the divisibility condition is off by a factor (ideally, it would only require t | n). All logarithms will be in base e ≈ 2.718

Concentration inequalities

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3