Two-disjoint-cycle-cover edge bipancyclicity of bipartite generalized hypercubes

Source abstract

Let \(G=C(d_1,\ldots,d_n)=F_1\BoxProd\cdots\BoxProd F_n\) be a bipartite generalized hypercube with $n\geq2$, all $d_i$ even, and $N=|V(G)|\geq8$, where $F_i=K_2$ when $d_i=2$, and $F_i=C_{d_i}$ when $d_i\geq4$. We prove the following exact strengthening of two-disjoint-cycle-cover vertex bipancyclicity. For every ordered pair of independent edges $e,f\in E(G)$ and every even integer $4\leq\ell\leq N-4$, the vertex set can be partitioned into two cycles $J_1,J_2$ of lengths $\ell$ and $N-\ell$, respectively, with $e\in E(J_1)$ and $f\in E(J_2)$, if and only if $G$ is not isomorphic to any \(K_2\BoxProd C_{2p}\) with $p\geq3$. The graph $C(2,2)\cong C_4$ is treated separately: it has no 2-DCC. Consequences that retain prescribed-edge information include ordinary edge bipancyclicity and the even $k$-ary $n$-cube specialization.