Let {Gi} be the random graph process: starting with an empty graph G0 with n vertices, in every step i ≥ 1 the graph Gi is formed by taking an edge chosen uniformly at random among the nonexisting ones and adding it to the graph Gi − 1. The classical “hitting‐time” result of Ajtai, Komlós, and Szemerédi, and independently Bollobás, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches 2, that is if δ(Gi) ≥ 2 then Gi is Hamiltonian. We establish a resilience version of this result. In particular, we show that the random graph process almost surely creates a sequence of graphs such that for m≥false(16+ofalse(1false)false)nnormallogn edges, the 2‐core of the graph Gm remains Hamiltonian even after an adversary removes false(12−ofalse(1false)false)‐fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.
The famous Pósa‐Seymour conjecture, confirmed in 1998 by Komlós, Sárközy, and Szemerédi, states that for any k ≥ 2, every graph on n vertices with minimum degree kn/(k + 1) contains the kth power of a Hamilton cycle. We extend this result to a sparse random setting. We show that for every k ≥ 2 there exists C > 0 such that if p≥Cfalse(normallognfalse/nfalse)1false/k then w.h.p. every subgraph of a random graph Gn,p with minimum degree at least (k/(k + 1) + o(1))np, contains the kth power of a cycle on at least (1 − o(1))n vertices, improving upon the recent results of Noever and Steger for k = 2, as well as Allen, Böttcher, Ehrenmüller, and Taraz for k ≥ 3. Our result is almost best possible in three ways: for p ≪ n−1/k the random graph Gn,p w.h.p. does not contain the kth power of any long cycle; there exist subgraphs of Gn,p with minimum degree (k/(k + 1) + o(1))np and Ω(p−2) vertices not belonging to triangles; there exist subgraphs of Gn,p with minimum degree (k/(k + 1) − o(1))np which do not contain the kth power of a cycle on (1 − o(1))n vertices.
A seminal result by Komlós, Sarközy, and Szemerédi states that if a graph G with n vertices has minimum degree at least kn{pk `1q, for some k P N and n sufficiently large, then it contains the k-th power of a Hamilton cycle. This is easily seen to be the largest power of a Hamilton cycle one can guarantee, given such a minimum degree assumption. Following a recent trend of studying effects of adding random edges to a dense graph, the model known as the randomly perturbed graph, Dudek, Reiher, Ruciński, and Schacht showed that if the minimum degree is at least kn{pk `1q `αn, for any constant α ą 0, then adding Opnq random edges on top almost surely results in a graph which contains the pk `1q-st power of a Hamilton cycle. We show that the effect of these random edges is significantly stronger, namely that one can almost surely find the p2k `1q-st power. This is the largest power one can guarantee in such a setting.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.