Local resilience of spanning subgraphs in sparse random graphs

Abstract

For each real γ > 0 and integers ∆ ≥ 2 and k ≥ 1, we prove that there exist constants β > 0 and C > 0 such that for all p ≥ C(log n/n) 1/∆ the random graph G(n, p) asymptotically almost surely contains -even after an adversary deletes an arbitrary (1/k − γ)-fraction of the edges at every vertex -a copy of every nvertex graph with maximum degree at most ∆, bandwidth at most βn and at least C max{p −2 , p −1 log n} vertices not in triangles.

Spanning universality in random graphs

“…is a threshold for the existence of a K Δ+1 -factor. It is believed that K Δ+1 -factors are graphs which are most "difficult" to embed, meaning that for any other graph H ∈ (n, Δ) the bound on p given in (1) suffices. This belief is supported by a result of Riordan [30] which shows that p ≥ n −2∕(Δ+1)+ (Δ) suffices, for some (Δ) > 0 which goes to 0 as Δ goes to infinity, and a recent result of Ferber, Luh and Nguyen [15] where they showed that this is indeed the case if H has at most (1 − )n vertices (we call such graphs almost-spanning).…”

“…This belief is supported by a result of Riordan [30] which shows that p ≥ n −2∕(Δ+1)+ (Δ) suffices, for some (Δ) > 0 which goes to 0 as Δ goes to infinity, and a recent result of Ferber, Luh and Nguyen [15] where they showed that this is indeed the case if H has at most (1 − )n vertices (we call such graphs almost-spanning). Moreover, it is also believed that for p being of order what is given in (1), not only that G n,p contains one such graph H ∈ (n, Δ) almost surely but it contains every graph from (n, Δ) simultaneously. In other words, a typical G n,p is (n, Δ)-universal.…”

“…Therefore, at least intuitively, one can expect to find a copy of any graph with maximum degree Δ by embedding it “vertex‐by‐vertex.” This bound has become a benchmark in the field and much subsequent work on embedding spanning or almost‐spanning graphs of maximum degree Δ in random graphs achieves this threshold. This includes the work of Kohayakawa, Rödl, Schacht and Szemerédi on Ramsey properties of random graphs, the bandwidth theorem for random graphs and a blow‐up lemma by Allen and coworkers . As all these results build on the ideas established for proving the above mentioned universality results, in order to achieve any further progress in these more involved questions we first need to improve our understanding of the universality problem.…”

Local resilience of an almost spanning k‐cycle in random graphs

“…For local resilience, the best previously known results are by Noever and Steger and Allen, Böttcher, Ehrenmüller, and Taraz , for the square of a cycle and higher powers, respectively. The former states that for any ε , γ > 0, if p ≥ N −1/2 + γ the local resilience of GN , p with respect to having the square of a cycle on (1 − ε ) N vertices is 1/3 + o (1), while the latter is a variant of the bandwidth theorem for sparse random graphs and implies that for any k ≥ 2, if p ≫ N −1/(2 k ) the local resilience of GN , p with respect to containing a k ‐cycle on all but O ( p −2 ) vertices, is 1/( k + 1) + o (1).…”

“…Note that the “leftover” given by Theorem is εN , which for k = 2 and the conjectured appearance threshold p ≥ N −1/2 would correspond to the optimal O ( p −2 ). However, as our bound on p is slightly larger, this leaves a small gap with respect to the result of Allen et al .…”