Resilience of perfect matchings and Hamiltonicity in random graph processes

Abstract

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 edges, the 2‐core of the graph Gm remains Hamiltonian even after an adversary removes ‐fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.