# Change Ringing and Hamiltonian Cycles : The Search for Erin and Stedman Triples

## TL;DR

This paper models the problem of finding specific bell-ringing sequences as constrained Hamiltonian cycle problems, introducing a method to convert complex ringing constraints into standard HCP instances, revealing their computational difficulty.

## Contribution

It presents a novel approach to encode campanology peal problems as Hamiltonian cycle problems and demonstrates how to decompose these into smaller, computationally challenging instances.

## Key findings

- Converted campanology constraints into standard HCP instances
- Partitioned complex problems into smaller, manageable subproblems
- Identified that known solution instances are exceptionally difficult for HCP algorithms

## Abstract

A very old problem in campanology is the search for peals. The latter can be thought of as a heavily constrained sequence of all possible permutations of a given size, where the exact nature of the constraints depends on which method of ringing is desired. In particular, we consider the methods of bobs-only Stedman Triples and Erin Triples; the existence of the latter is still an open problem. We show that this problem can be viewed as a similarly constrained form of the Hamiltonian cycle problem (HCP). Through the use of special subgraphs, we convert this to a standard instance of HCP. The original problem can be partitioned into smaller instances, and so we use this technique to produce smaller instances of HCP as well. We note that the instances known to have solutions provide exceptionally difficult instances of HCP.