# Hamiltonicity in random directed graphs is born resilient

## TL;DR

This paper proves that in a random directed graph process, once a certain minimum degree is achieved, the graph is almost surely resiliently Hamiltonian, maintaining Hamiltonicity despite significant edge removals, improving previous resilience bounds.

## Contribution

It establishes the almost sure resilience threshold for Hamiltonicity in the random directed graph process, improving previous results on resilience in random directed graphs.

## Key findings

- Directed graphs with minimum degree at least 1 are resiliently Hamiltonian up to nearly half edge removal.
- Almost surely, the random directed graph process does not achieve resilience beyond half edge removal.
- The results improve upon prior bounds for resilience in random directed graphs.

## Abstract

Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $(1/2-\varepsilon)$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each $\varepsilon>0$, we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$-resiliently Hamiltonian. This improves a result of Ferber, Nenadov, Noever, Peter and \v{S}kori\'{c}, who showed, for each $\varepsilon>0$, that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$-resiliently Hamiltonian if $p=\omega(\log^8n/n)$.