Spanning universality in random graphs

TL;DR

This paper improves the known threshold for the universality of random graphs in containing all graphs with bounded maximum degree, advancing the understanding of embedding spanning structures.

Contribution

It demonstrates that the random graph $G_{n,p}$ is almost surely universal for a lower threshold $p = ilde Omega(n^{-1/(elta-1/2)})$, surpassing previous bounds.

Findings

Improved the universality threshold for random graphs.

Extended embedding techniques beyond previous limitations.

Achieved near-optimal conditions for spanning structure embedding.

Abstract

A graph is said to be $\mathcal{H}(n, \Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. Using a `matching-based' embedding technique introduced by Alon and F\"uredi, Dellamonica, Kohayakawa, R\"odl and Ruci\'nski showed that the random graph $G_{n,p}$ is asymptotically almost surely $\mathcal{H}(n, \Delta)$-universal for $p = \tilde \Omega(n^{-1/\Delta})$ - a threshold for the property that every subset of $\Delta$ vertices has a common neighbour. This bound has become a benchmark in the field and many subsequent results on embedding spanning structures of maximum degree $\Delta$ in random graphs are proven only up to this threshold. We take a step towards overcoming limitations of former techniques by showing that $G_{n,p}$ is almost surely $\mathcal{H}(n, \Delta)$-universal for $p = \tilde \Omega(n^{- 1/(\Delta-1/2)})$.