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 $gcd(dm,cn)=2$.