4-regular 4-connected Hamiltonian graphs with few Hamiltonian cycles

TL;DR

This paper constructs an infinite family of 4-regular, 4-connected Hamiltonian graphs with a limited number of Hamiltonian cycles, highlighting structural properties of such graphs.

Contribution

It introduces a new family of graphs with specific regularity and connectivity properties that have few Hamiltonian cycles, advancing understanding of Hamiltonian graph structures.

Findings

Existence of an infinite family of 4-regular 4-connected Hamiltonian graphs with few Hamiltonian cycles

Open question about similar families for 5-regular 5-connected Hamiltonian graphs

Bounded number of Hamiltonian cycles in the constructed graphs

Abstract

We prove that there exists an infinite family of 4-regular 4-connected Hamiltonian graphs with a bounded number of Hamiltonian cycles. We do not know if there exists such a family of 5-regular 5-connected Hamiltonian graphs.