Cayley digraphs and (1, j, n)-sequencings of the alternating groups An

Abstract

The problem of finding a sequencing Π 1, Π 2,… Π | A n | for the elements of the alternating group A n which minimizes the cost function C= ∑ i=1 |A n|−1 c(П i −1∘П i+1) where c( Π) = |{ j: Π( j) ≠ j}|, is solved. For n ⩾ 3 the sequencing is constructed by finding a directed Hamiltonian path in the Cayley digraph D n of A n , dtermined by the generating set B n = {(1, j, n): j ϵ { 2,3,…, n−1}}. We further consider the question of finding a directed Hamiltonian cycle in D n .