# Finding Hamilton cycles in random intersection graphs

## TL;DR

This paper investigates the problem of finding Hamilton cycles in random intersection graphs, analyzing a classical algorithm's efficiency and establishing threshold functions for successful cycle construction.

## Contribution

It demonstrates that the classical algorithm HAM has a threshold for Hamilton cycle detection in G(n,m,p) matching the minimum degree threshold, extending understanding beyond previous algorithms.

## Key findings

- Threshold for HAM matches minimum degree at least two
- Algorithm HAM is effective in broader parameter ranges
- Provides new insights into Hamilton cycle detection in intersection graphs

## Abstract

The construction of the random intersection graph model is based on a random family of sets. Such structures, which are derived from intersections of sets, appear in a natural manner in many applications. In this article we study the problem of finding a Hamilton cycle in a random intersection graph. To this end we analyse a classical algorithm for finding Hamilton cycles in random graphs (algorithm HAM) and study its efficiency on graphs from a family of random intersection graphs (denoted here by G(n,m,p)). We prove that the threshold function for the property of HAM constructing a Hamilton cycle in G(n,m,p) is the same as the threshold function for the minimum degree at least two. Until now, known algorithms for finding Hamilton cycles in G(n,m,p) were designed to work in very small ranges of parameters and, unlike HAM, used the structure of the family of random sets.