Cycles in enhanced hypercubes

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