The global resilience of Hamiltonicity in $G(n, p)$

TL;DR

This paper investigates the resilience of Hamiltonicity in random graphs, showing that above the threshold, the minimal edge removal to destroy Hamiltonicity is typically the minimum degree minus one.

Contribution

It establishes the exact value of the global resilience of Hamiltonicity in Erdős–Rényi random graphs across all edge probabilities.

Findings

Resilience equals minimum degree minus one above the threshold

Resilience equals max of zero and minimum degree minus one for all p

High probability results for the global resilience in G(n,p)

Abstract

Denote by $r_g(G,\mathcal{H})$ the global resilience of a graph $G$ with respect to Hamiltonicity. That is, $r_g(G,\mathcal{H})$ is the minimal $r$ for which there exists a subgraph $H\subseteq G$ with $r$ edges, such that $G\setminus H$ is not Hamiltonian. We show that if $p$ is above the Hamiltonicity threshold and $G\sim G(n,p)$ then, with high probability, $r_g(G,\mathcal{H})=\delta (G)-1$. This is easily extended to the full interval: for every $p(n)\in [0,1]$, if $G\sim G(n,p)$ then, with high probability, $r_g(G,\mathcal{H})= \max \{ 0,\delta (G)-1 \}$.