Resilience of Perfect Matchings and Hamiltonicity in Random Graph Processes

TL;DR

This paper proves that in a random graph process, the 2-core remains Hamiltonian and contains perfect matchings despite significant adversarial edge removals, demonstrating resilience properties of these structures.

Contribution

It establishes a resilience version of the hitting-time result for Hamiltonicity and perfect matchings in the random graph process.

Findings

The 2-core remains Hamiltonian after removing nearly half of incident edges.

Hamiltonian properties are resilient to adversarial edge deletions.

Results hold for graphs with about (1/6) n log n edges.

Abstract

Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ 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, Koml\'{o}s, and Szemer\'{e}di, and independently Bollob\'{a}s, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches $2$, that is if $\delta(G_i) \ge 2$ 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 $m \geq (\tfrac{1}{6} + o(1))n\log n$ edges, the $2$-core of the graph $G_m$ remains Hamiltonian even after an adversary removes $(\tfrac{1}{2} - o(1))$-fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.