Resilience of Perfect Matchings and Hamiltonicity in Random Graph Processes [article]

Rajko Nenadov, Angelika Steger, Miloš Trujić
2018 arXiv   pre-print
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 non-existing 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.
more » ... 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 ≥ (16 + o(1))n n edges, the 2-core of the graph G_m remains Hamiltonian even after an adversary removes (12 - o(1))-fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.
arXiv:1710.00799v3 fatcat:ldm7qiptwjhopjpnyghvvzhqma