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

Abstract

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 = Z 8 m ⋊ Z 2 n = 〈 x , y : x 8 m = 1 , y 2 n = 1 , and yxy - 1 = x 4 m + 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 ) , ( 4 mn 2 + 2 n , 16 m 2 n ) , ( n , 4 m ) , and ( 0 , 8 m ) . Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by G = Z 4 m ⋊ Z 2 n = 〈 x , y : x 4 m = 1 , y 2 n = 1 , and yxy - 1 = x 2 m - 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is ( 3 m - 1 ) n + m ⌊ ( n + 1 ) / 3 ⌋ + 1 .