Orbits of Hamiltonian Paths and Cycles in Complete Graphs

TL;DR

This paper counts geometric equivalence classes of Hamiltonian paths and cycles in complete graphs using group actions and Burnside's lemma, providing new enumeration methods and alternative proofs for known formulas.

Contribution

It introduces a novel approach to enumerate orbits of Hamiltonian subgraphs via group actions, offering an alternative proof of existing enumeration formulas.

Findings

Enumerates orbits of Hamiltonian paths and cycles under group actions.

Provides an alternative proof for known enumeration formulas.

Uses Burnside's lemma for counting geometric equivalence classes.

Abstract

We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings representing subgraphs. These orbits are enumerated using Burnside's lemma. The technique used also provides an alternative proof of the formulae found by S. W. Golomb and L. R. Welch which give the number of distinct $n$-gons on fixed, regularly spaced vertices up to rotation and optionally reflection.