Resilience of perfect matchings and Hamiltonicity in random graph processes. Issue 4 (8th December 2018) - Nenadov, Rajko; Steger, Angelika; Trujić, Miloš
Abstract : Let { G i } be the random graph process: starting with an empty graph G 0 with n vertices, in every step i  ≥ 1 the graph G i is formed by taking an edge chosen uniformly at random among the nonexisting ones and adding it to the graph G i  − 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 ( G i ) ≥ 2 then G i 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 ≥ ( 1 6 + o ( 1 ) ) n log n edges, the 2‐core of the graph G m remains Hamiltonian even after an adversary removes ( 1 2 − o ( 1 ) ) ‐fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.