We obtain a characterization of the Hamilton paths in the cartesian product Z x Zb of two directed cycles. This provides a correspondence between the collection of Hamilton paths in Za x Zb and the set of visible lattice points in the triangle with vertices (0,O) , (0,a) , and (b,0) . We use this correspondence to show there is a Hamilton circuit in the cartesian product of any three or more nontrivial directed cycles. Our methods are a synthesis of the theory of torus knots and the study of Hamilton paths in Cayley digraphs of abelian groups.
Suppose G is a finite group, such that |G| = 16p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G; S).
For a graph G = (V, E) naturally embedded in the torus, let F(G) denote the set of faces of G. Then, G is called a C n -face-magic toroidal graph if there exists a bijection f : V (G) → {1, 2, . . . , |V (G)|} such that for every F ∈ F(G) with F ∼ = C n , the sum of all the vertex labels along C n is a constant S. LetWe show that, for all m, n ≥ 2, C m × C n admits a C 4 -face-magic toroidal labeling if and only if either m = 2, or n = 2, or both m and n are even. We say that a C 4 -face-magic toroidal labeling {x i,j :We show that there exists an antipodal balanced C 4 -facemagic toroidal labeling on C 2m × C 2n if and only if the parity of m and n are the same. Furthermore, when both m and n are even, an antipodal balanced C 4 -face-magic toroidal labeling on C 2m × C 2n is both row-sum balanced and column-sum balanced. In addition, when m = n is even, an antipodal balanced C 4 -face-magic toroidal labeling on C 2n × C 2n is diagonal-sum balanced.
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