Hamiltonian cycles in Cayley graphs whose order has few prime factors

TL;DR

This paper proves that Cayley graphs with a number of vertices having a limited prime factorization always contain a Hamiltonian cycle, extending known results to new classes of graphs with specific prime factorizations.

Contribution

It establishes Hamiltonian cycle existence in Cayley graphs where the order's prime factorization is restricted to certain small forms, broadening previous understanding.

Findings

Hamiltonian cycles exist in Cayley graphs with orders of specific prime factorizations.

Results cover orders with up to three prime factors and certain powers of primes.

The paper extends known conditions for Hamiltonicity in Cayley graphs.

Abstract

We prove that if Cay(G;S) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay(G;S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with k < 32 and k not equal to 24, or of the form kpq with k < 6, or of the form pqr, or of the form kp^2 with k < 5, or of the form kp^3 with k < 3.