Hamiltonian Cycles in Subdivided Doubles

TL;DR

This paper investigates subdivided double constructions of 4-regular graphs, revealing they contain exponentially many pairs of complementary Hamiltonian cycles, regardless of their symmetry properties.

Contribution

It demonstrates that subdivided doubles have exponentially many Hamiltonian cycles, each complementary to another, expanding understanding of their cycle structure beyond symmetry considerations.

Findings

Subdivided doubles have exponentially many Hamiltonian cycles.

Each Hamiltonian cycle has a complementary cycle.

This property holds regardless of the graph's symmetry.

Abstract

The subdivided double construction on 4-regular graphs was used by Poto\v{c}nik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that contains a pair of twin vertices. We show that (regardless of symmetry) subdivided doubles have another curious property: they have exponentially many Hamiltonian cycles each of which is complementary to another Hamiltonian cycle.