Hamiltonicity of the Cayley Digraph on the Symmetric Group Generated by {\sigma} = (1 2 ... n) and {\tau} = (1 2)

TL;DR

This paper constructs Hamilton paths and cycles in the directed Cayley graph of the symmetric group generated by specific permutations, solving an open problem for all n and for odd n respectively.

Contribution

It provides explicit constructions of Hamilton paths for all n and Hamilton cycles for odd n in the Cayley graph of the symmetric group, addressing an open problem.

Findings

Hamilton paths constructed for all n

Hamilton cycles constructed for odd n

Solved an open problem by Nijenhuis and Wilf

Abstract

The symmetric group is generated by {\sigma} = (1 2 ... n) and {\tau} = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n.