Almost-spanning universality in random graphs

TL;DR

This paper improves the understanding of universality in random graphs, showing that for maximum degree , the threshold for containing all such graphs can be lowered significantly beyond the natural boundary.

Contribution

It demonstrates that for , the universality threshold in random graphs is lower than previously known, surpassing the natural boundary with a new probabilistic bound.

Findings

For , universality holds at p = (n^{-rac{1}{\u0003-1}} ext{log}^5 n)

Improves previous thresholds for universality in random graphs

Shows universality for a broader class of graphs with higher probability

Abstract

A graph $G$ is said to be $\mathcal H(n,\Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. It is known that for any $\varepsilon > 0$ and any natural number $\Delta$ there exists $c > 0$ such that the random graph $G(n,p)$ is asymptotically almost surely $\mathcal H((1-\varepsilon)n,\Delta)$-universal for $p \geq c (\log n/n)^{1/\Delta}$. Bypassing this natural boundary, we show that for $\Delta \geq 3$ the same conclusion holds when $p = \omega\left(n^{-\frac{1}{\Delta-1}}\log^5 n\right)$.