Symmetry classes of Hamiltonian cycles

TL;DR

This paper explores Hamiltonian cycles under graph symmetries, focusing on Hamiltonian-transitive graphs, and shows that certain Cayley graphs are not Hamiltonian-transitive, providing new constructions and insights.

Contribution

It introduces the concept of Hamiltonian-transitivity, proves that Cayley graphs of abelian groups are not Hamiltonian-transitive under mild conditions, and constructs infinite families of such graphs.

Findings

Cayley graphs of abelian groups are not Hamiltonian-transitive under certain conditions.

Hamiltonian-transitivity relates to prime factors of Cartesian product decompositions.

Constructed infinite families of regular Hamiltonian-transitive graphs.

Abstract

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.