Change ringing and Hamiltonian cycles: The search for Erin and Stedman triples

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 (but not previously considered) 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. Introduction and TerminologyChange ringing is the art of striking a set of bells in constantly varying orders, such that at each stage, each bell is struck once before the order changes. Typically, each bell has a designated person, or ringer, assigned to strike it, and at each stage, the order in which the ringers strike the bells depends on various schemes. Such an order is called a row, and initially, the bells are struck in decreasing order of the tone they produce; this row is called rounds. Each row can be represented as a sequence of n numbers (assuming there are n bells) where the first number denotes which bell should be struck first, the second number denotes which bell should be struck second, and so on. The bells are labelled in order from the highest tone to the lowest. For example, suppose there are five bells; then rounds could be represented as 12345. In this notation, we say that the first bell is in the first place, and so on.

FHCP Challenge Set: The First Set of Structurally Difficult Instances of the Hamiltonian Cycle Problem

“…The sixteen graphs not solved by Cohen and Coudert were not solved in any of the submissions. These sixteen instance comprised three instances from [27], namely Sted4 (6930 vertices), Sted5 (5544 vertices) and Sted6 (4620 vertices), as well as thirteen instances made out of combinations of graphs which included at least one of Sted4, Sted5 and Sted6. Of all the instances included in the FHCP Challenge Set, these instances were by far the most resistant to attack from heuristics.…”

“…Of all the instances included in the FHCP Challenge Set, these instances were by far the most resistant to attack from heuristics. It should be noted that two other instances from [27] were included, namely Sted10 (2772 vertices) and Sted20 (1386 vertices). Of these, Sted10 was only solved by Cohen and Coudert, and Sted20 was only solved by Johnson, and Cohen and Coudert.…”

“…In general, it seemed that SLH and LKH struggled to solve these instances, with Concorde successful for some but not others. Finally, instances of HCP which were recently derived from a famous combinatorial problem from campanology, specifically the search for peals in parts of bobs-only Stedman Triples [27], were also included; none of the algorithms tested were able to easily solve these instances.…”

On hamiltonicity of 1-tough triangle-free graphs

“…Hamiltonicity has been a central topic in structural graph theory since the 1950s, and has regained more popularity since the development of algorithmic graph theory and the discovery that the associated decision problem is NP-complete. It also has many and sometimes very surprising applications, as the recent publication [5] in this journal shows. Forbidden subgraph conditions for guaranteeing hamiltonian properties have been studied intensively since the PhD thesis of Bedrossian [1] appeared in 1991.…”

A construction for directed in-out subgraphs of optimal size

“…We consider a family of graphs which possess a property called in-out, recently defined in Haythorpe and Johnson [7]. Interchangably, they may be referred to as in-out graphs or in-out subgraphs, with the latter name used because many applications occur when they are included as part of a larger graph.…”