Spanning universality in random graphs

A graph is said to be H(n,Delta) -universal if it contains every graph with n vertices and maximum degree at most Delta as a subgraph. Dellamonica, Kohayakawa, Rodl and Rucinski used a "matching-based" embedding technique introduced by Alon and Furedi to show that the random graph Gn,p is asymptotically almost surely H(n,Delta) -universal for p=omega((logn/n)1/Delta), a threshold for the property that every subset of Delta vertices has a common neighbor. This bound has become a benchmark in the field and many subsequent results on embedding spanning graphs 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 Gn,p is almost surely H(n,Delta) -universal for p=omega(n-1/(Delta-0.5)log3n).