The global resilience of Hamiltonicity 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, $r_g(G,\mathcal{H})= \max \{ 0,\delta (G)-1 \}$.