Cycles in enhanced hypercubes

The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length from $4$ to $2^n$; if $k$ is even, every edge of $Q_{n,k}$ also lies on a cycle of every odd length from $k+3$ to $2^n-1$, and some special edges lie on a shortest odd cycle of length $k+1$.

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