Hamilton Cycles in Restricted Rotator Graphs

Abstract

The rotator graph has vertices labeled by the permutations of n in one line notation, and there is an arc from u to v if a prefix of u’s label can be rotated to obtain v’s label. In other words, it is the directed Cayley graph whose generators are $\sigma_{k} := (1 \ 2 \ \cdots \ k)$ for 2 ≤ k ≤ n and these rotations are applied to the indices of a permutation. In a restricted rotator graph the allowable rotations are restricted from k ∈ {2,3,…,n} to k ∈ G for some smaller (finite) set G ⊆ {2,3,…,n}. We construct Hamilton cycles for G = {n−1,n} and G = {2,3,n}, and provide efficient iterative algorithms for generating them. Our results start with a Hamilton cycle in the rotator graph due to Corbett (IEEE Transactions on Parallel and Distributed Systems 3 (1992) 622–626) and are constructed entirely from two sequence operations we name ‘reusing’ and ‘recycling’.