Uniquely hamiltonian graphs for many sets of degrees

TL;DR

This paper provides constructive methods to prove the existence of uniquely hamiltonian graphs with various degree sets, advancing understanding of graph structures and their connectivity properties.

Contribution

It introduces new constructions for uniquely hamiltonian graphs across multiple degree sets and the concept of seeds, with implications for Sheehan's conjecture.

Findings

Constructed uniquely hamiltonian graphs for all degree sets with minimum 2 or 3 containing an even number.

Established the existence of 3-connected uniquely hamiltonian graphs for degrees 3 and 4.

Proved the equivalence of existence between 3-connected and 2-connected uniquely hamiltonian 4-regular graphs.

Abstract

We give constructive proofs for the existence of uniquely hamiltonian graphs for various sets of degrees. We give constructions for all sets with minimum 2 (a trivial case added for completeness), all sets with minimum 3 that contain an even number (for sets without an even number it is known that no uniquely hamiltonian graphs exist), and all sets with minimum 4, except {4}, {4,5}, and {4,6}. For minimum degree 3 and 4, the constructions also give 3-connected graphs. We also introduce the concept of seeds, which makes the above results possible and might be useful in the study of Sheehan's conjecture. Furthermore, we prove that 3-connected uniquely hamiltonian 4-regular graphs exist if and only if 2-connected uniquely hamiltonian 4-regular graphs exist.