Spanning universality in random graphs

Abstract

A graph is said to be H(n,Δ) ‐universal if it contains every graph with n vertices and maximum degree at most Δ as a subgraph. Dellamonica, Kohayakawa, Rödl and Ruciński used a “matching‐based” embedding technique introduced by Alon and Füredi to show that the random graph Gn,p is asymptotically almost surely H(n,Δ) ‐universal for p=Ω((logn/n)1/Δ), a threshold for the property that every subset of Δ 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 Δ 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,Δ) ‐universal for p=Ω(n−1/(Δ−0.5)log3n).

Random Perturbation of Sparse Graphs

“…Similar as above there was a tremendous amount of research on determining the thresholds for various spanning structures, e.g. for matchings [17], trees [32,36], factors [24], powers of Hamilton cycles [35,37], and general bounded degree graphs [1,18,19,40]. An extensive survey by Böttcher can be found in [9].…”

Clique-factors in sparse pseudorandom graphs

“…[14,Lemma 2.8]. The use of a sparse template as part of the absorbing method was also applied by Kwan in [13], where he generalized the notion to 3-uniform hypergraphs in order to study random Steiner triple systems, and by Ferber and Nenadov [6] in their work on the universality of random graphs.…”

Finding any given 2‐factor in sparse pseudorandom graphs efficiently

“…Kwan [40] also used sparse templates to study random Steiner triple systems, generalising the template to a hypergraph setting and using it to define an absorbing structure for perfect matchings. Further applications were given by Ferber and Nenadov [23] in their work on universality in the random graph, recently by the current authors in [24] which was the first use of the method in the context of pseudorandom graphs, and by Nenadov and Pehova [45] who used the method to study a variant of the Hajnal‐Szeméredi Theorem. The final three papers mentioned all adapt the method to give absorbing structures which output disjoint copies of a fixed graph (a partial ‐factor), however the different absorbing structures used are interestingly all significantly distinct.…”

“…In this case we say an ‐vertex graph is ‐universal if it contains all graphs on at most vertices of maximum degree . A large part of the focus of the study has been on the universality properties of [9,18,20,22,23,29]. It is also natural to investigate the universality properties of ‐graphs as was suggested by Krivelevich, Sudakov and Szabó in [38].…”