Hamilton Cycles in Random Lifts of Graphs

TL;DR

This paper proves that random lifts of graphs with minimum degree at least 5 and two disjoint Hamiltonian cycles are almost surely Hamiltonian, expanding understanding of Hamiltonicity in random graph constructions.

Contribution

It establishes conditions under which random lifts of graphs are almost surely Hamiltonian, specifically when the base graph has minimum degree 5 and two disjoint Hamiltonian cycles.

Findings

Random lifts of certain graphs are almost surely Hamiltonian.

Minimum degree 5 and specific cycle conditions ensure Hamiltonicity.

Provides probabilistic proof for Hamiltonian cycles in lifted graphs.

Abstract

For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that asymptotically almost surely the random lift of a graph $G$ is hamiltonian, provided $G$ has the minimum degree at least $5$ and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.