2-universality in randomly perturbed graphs

Abstract

A graph G is called universal for a family of graphs F if it contains every element F P F as a subgraph. Let Fpn, 2q be the family of all graphs with maximum degree 2. Ferber, Kronenberg, and Luh [Optimal Threshold for a Random Graph to be 2-Universal, to appear in Transactions of the American Mathematical Society] proved that there exists a C such that for p ě Cpn´2 {3 log 1{3 nq the random graph Gpn, pq a.a.s is Fpn, 2q-universal, which is asymptotically optimal. For any n-vertex graph Gα with minimum degree δpGαq ě αn Aigner and Brandt [Embedding arbitrary graphs of maximum degree two, Journal of the London Mathematical Society 48 (1993), 39-51] proved that Gα is Fpn, 2q-universal for an optimal α ě 2{3.In this note, we consider the model of randomly perturbed graphs, which is the union GαYGpn, pq. We prove that a.a.s. Gα Y Gpn, pq is Fpn, 2q-universal provided that α ą 0 and p " ωpn´2 {3 q. This is asymptotically optimal and improves on both results from above in the respective parameter. Furthermore, this extends a result of Böttcher, Montgomery, Parczyk, and Person [Embedding spanning bounded degree subgraphs in randomly perturbed graphs, arXiv:1802.04603 (2018)], who embed a given F P Fpn, 2q at these values. We also prove variants with universality for the family F ℓ pn, 2q, all graphs from Fpn, 2q with girth at least ℓ. For example, there exists an ℓ0 depending only on α such that for all ℓ ě ℓ0 already p " ωp1{nq is sufficient for F ℓ pn, 2q-universality.but the extra log n-term ist needed to guarantee that the graph has a.a.s. minimum degree 2. In the following discussion we will work with p " ωppq even though for many results a stronger variant is proved, where p ě Cp for some absolute constant C depending only on pF n q ně1 .1.1. Randomly perturbed graphs. Combining the two models from random and extremal graph theory, Bohman, Frieze, and Martin [7] introduced the model of randomly perturbed graphs G α Y Gpn, pq for any α ą 0, where, as above, G α is any n-vertex graph with minimum degree δpG α q ě αn. They show that p " ωp1{nq is sufficient to a.a.s. guarantee a Hamilton cycle in G α Y Gpn, pq for any G α . When G α is the complete unbalanced bipartite graph K αn,p1´αqn then at least a linear number of egdes is needed. Using both graphs G α and Gpn, pq this result dramatically improves on the α ě 1{2 needed in G α alone, even though adding Gpn, pq is a relatively small random perturbation. On the other hand it is also a log n-term better than the threshold in Gpn, pq alone, which is plausible as G α guarantees a large minimum degree.In recent years this model attracted a lot of attention. For a bounded degree spanning tree Krivelevich, Kwan, and Sudakov [25] proved that p " ωp1{nq also is sufficient in G α Y Gpn, pq. In G α alone α ą 1{2 is needed [22] and in Gpn, pq only recently Montgomery [28] was able to show that again log n{n gives the threshold. Similar results were proved for factors in [3] and for powers of Hamilton cycles and general bounded degree graphs in [10]. Together with Böttcher,...

The square of a Hamilton cycle in randomly perturbed graphs

“…The range 1 3 ≤ α < 2 3 is a corollary of our Theorem 1.1, while the remaining cases follow from known results. For α = 0, this is due to Ferber, Kronenberg, and Luh [14], for α ≥ 2 3 to Aigner and Brandt [1], and for α ∈ (0, 1 3 ) to [28]. The result in the range α ∈ [ 1 3 , 2 3 ) significantly strengthens one of our results from [7], where the same result was established for the containment of a triangle factor only.…”

Triangles in randomly perturbed graphs

“…Even further, we call a graph 2-universal if it contains any n-vertex graph of maximum degree 2 as a subgraph. It is known that the threshold for 2universality is asymptotically the same as for a K 3 -factor when α < 1 3 or α ≥ 2 3 [1,14,27]. In a follow up paper we will expand our approach and prove that this also holds for the remaining cases, i.e.…”

Triangles in randomly perturbed graphs

“…Even further, we call a graph 2-universal if it contains any n-vertex graph of maximum degree 2 as a subgraph. It is known that the threshold for 2-universality is asymptotically the same as for a triangle factor when α < 1/3 or α ≥ 2/3 [1,15,28]. In a follow-up paper, we will expand our approach and prove that this also holds for the remaining cases, i.e.…”