Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd(m, n) = 1. Let G be the semidirect product of cyclic groups given by G= Z8m Z2n = 〈x, y : x8m = 1, y2n = 1, and yxy−1 = x4m+1〉. Then the number of hamilton paths in Cay(G : x, y) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are (0, 0), (4mn2+ 2n, 16m2n), (n, 4m), and (0, 8m). Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given byG=Z4m Z2n=〈x, y : x4m= 1, y2n= 1, and yxy−1= x2m−1〉. Then the number of hamilton paths in Cay(G : x, y) (with initial vertex 1) is (3m− 1)n+m (n+ 1)/3 + 1. © 2007 Elsevier B.V. All rights reserved.