Two-disjoint-cycle-cover edge bipancyclicity in Cartesian products

Abstract

For two even integers $r_1\le r_2$, a graph $G$ of even order is \emph{2-DCC edge $[r_1,r_2]$-bipancyclic} if for every ordered pair $(e,f)$ of independent edges and every even integer $\ell \in [r_1,r_2]$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ with $V(C_1)\cup V(C_2)=V(G)$, $|C_1|=\ell$, $|C_2|=|V(G)|-\ell$, $e\in E(C_1)$, and $f\in E(C_2)$. We study this property for Cartesian products $G=H_1\square \cdots \square H_d$ with $d\ge 2$, where each factor is $K_2$ or an even-order bipartite edge-Hamiltonian graph. We prove that $G$ is 2-DCC edge $[4,|V(G)|/2]$-bipancyclic whenever $d\ge 4$, and also for $d\in \{2,3\}$ under explicit Hamiltonian-cycle conditions on $H_i$. Here $1\le i\le d$. As applications, we give sharp characterizations for undirected toroidal meshes, star-cube networks, and hyperstar networks.