On the local resilience of random geometric graphs with respect to connectivity and long cycles

Given an increasing graph property P, a graph G is -resilient with respect to P if, for every spanning subgraph H G where each vertex keeps more than a (1-) -proportion of its neighbours, H has property P. We study the above notion of local resilience with G being a random geometric graph Gd (n, r) obtained by embedding n vertices independently and uniformly at random in 0, 1ᵈ, and connecting two vertices by an edge if the distance between them is at most r. First, we focus on connectivity. We show that, for every >0, for r a constant factor above the sharp threshold for connectivity rc of Gd (n, r), the random geometric graph is (1/2-) -resilient for the property of being k-connected, with k 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 r a constant factor above rc, Gd (n, r) is (1/2-) -resilient with respect to containing cycles of all lengths between constant and 2n/3. Proving (1/2-) -resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that Gd (n, r) is -resilient with respect to Hamiltonicity for a fixed constant = (d) <1/2.