Hyperhamiltonicity of The Cartesian Product of Two Directed Cycles

Abstract

Let \(G\) be the product of two directed cycles, let \(Z_a\) be a subgroup of \(Z_a\), and let \(Z_d\) be a subgroup of \(Z_b\). Also, let \(A = \frac{a}{c}\) and \(B = \frac{b}{d}\). We say that \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if there is a spanning connected subgraph of \(G\) that has degree \((2, 2)\) at the vertices of \(Z_c \times Z_d\) and degree \((1, 1)\) everywhere else. We show that the graph \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if and only if there exist positive integers \(m\) and \(n\) such that \(Am + Bn = AB + 1\), \(gcd(m, n) = 1\) or \(2\), and when \(gcd(m, n) = 2\), then \(gcd(dm, cn) = 2\).