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Abstract

A. Given an increasing graph property P, a graph is -resilient with respect to P if, for every spanning subgraph ⊆ where each vertex keeps more than a (1 − )-proportion of its neighbours, has property P. We study the above notion of local resilience with being a random geometric graph ( , ) obtained by embedding vertices independently and uniformly at random in [0, 1] , and connecting two vertices by an edge if the distance between them is at most .First, we focus on connectivity. We show that, for every > 0, for a constant factor above the sharp threshold for connectivity of ( , ), the random geometric graph is (1/2 − )-resilient for the property of being -connected, with of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small > 0, connectivity is not born (1/2 − )-resilient in 2-dimensional random geometric graphs.Second, we study local resilience with respect to the property of containing long cycles. We show that, for a constant factor above , ( , ) is (1/2 − )-resilient with respect to containing cycles of all lengths between constant and 2 /3. Proving (1/2 − )-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that ( , ) is -resilient with respect to Hamiltonicity for a fixed constant = ( ) < 1/2. This result can be regarded as a version of Dirac's theorem for binomial random graphs. Recently, Montgomery [38] and Nenadov, Steger and Trujić [42] independently sharpened this result to a "hitting time" result. Moreover, Theorem 1.1 was recently generalised by Condon, Espuny Díaz, Kim, Kühn and Osthus [14], who considered a version of the problem where the adversary is allowed to delete more edges incident to some, but not all, vertices of the graph, giving an analogue of a classical result of Pósa [50] for random graphs.The local resilience with respect to Hamiltonicity has also been studied in other models of random graphs. One example is the model of random regular graphs. Given integers 1 ≤ < , a random -regular graph , is obtained by sampling an element from the set of all -vertex -regular graphs uniformly at random. It is well known that the random -regular graph is a.a.s. Hamiltonian for all ≥ 3 [16, 35, 51, 52]. Improving on earlier work of Ben-Shimon, Krivelevich and Sudakov [8], Condon, Espuny Díaz, Girão, Kühn and Osthus [13] proved a result analogous to Theorem 1.1 for random regular graphs.Theorem 1.2 ([13, Theorem 1.2]). For every ∈ (0, 1/2], there exists a constant = ( ) > 0 such that, for all ∈ [ , − 1], a.a.s. every (1/2 + )-subgraph of , is Hamiltonian.Just like for binomial random graphs, the constant 1/2 is best possible. Moreover, Condon, Espuny Díaz, Girão, Kühn and Osthus [13] also showed that a polynomial dependency between and in Theorem 1.2 cannot be avoided.1 We remark that our definition of local resilience is slightly different from that in [55] and closer to more recent versions which have appeared in the literature [38, 42]. Nevertheless, all the results we mention in our paper hold with ...