# Dirac's theorem for random regular graphs

## TL;DR

This paper proves a resilience version of Dirac's theorem for random regular graphs, showing that large minimum degree subgraphs are Hamiltonian, confirming a conjecture and establishing optimal bounds.

## Contribution

It establishes the first resilience result for Hamiltonicity in random regular graphs, confirming a conjecture and proving optimal bounds.

## Key findings

- Subgraphs with minimum degree above half are Hamiltonian a.a.s.
- Resilience bounds are tight and cannot be improved.
- Results apply to sufficiently large degree d.

## Abstract

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.