An explicit universal cycle for the (n-1)-permutations of an n-set

TL;DR

This paper presents an explicit, efficient algorithm for constructing a Hamilton cycle in a directed Cayley graph related to permutations, fulfilling a long-standing open problem for explicit cycle construction.

Contribution

The authors provide a simple recursive method and a loopless algorithm to explicitly generate the Hamilton cycle in the permutation graph.

Findings

The cycle can be generated by a recursive algorithm.

The algorithm uses O(n) space and produces each edge in constant time.

It provides an explicit construction previously only known to exist.

Abstract

We show how to construct an explicit Hamilton cycle in the directed Cayley graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >... k). The existence of such cycles was shown by Jackson (Discrete Mathematics, 149 (1996) 123-129) but the proof only shows that a certain directed graph is Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and Permutations (2005)) asks for an explicit construction. We show that a simple recursion describes our Hamilton cycle and that the cycle can be generated by an iterative algorithm that uses O(n) space. Moreover, the algorithm produces each successive edge of the cycle in constant time; such algorithms are said to be loopless.