Cycles in cube-connected cycles graphs

Let nn be a positive integer with n≥3n≥3. The cube-connected cycles graph CCCnCCCn has n×2nn×2n vertices, labeled (l,x), where 0≤l≤n−10≤l≤n−1 and x is an nn-bit binary string. Two vertices (l,x) and (l′,y) are adjacent if and only if either x=y and |l−l′|=1|l−l′|=1, or l=l′l=l′ and y=(x)l. Let L(n)L(n) denote the set of all possible lengths of cycles in CCCnCCCn. In this paper, we prove that L(n)={n}∪{i∣iL(n)={n}∪{i∣i is even, 8≤i≤n+58≤i≤n+5, and i≠10}∪{i∣n+6≤i≤n2n}i≠10}∪{i∣n+6≤i≤n2n} if nn is odd; L(4)={4}∪{i∣iL(4)={4}∪{i∣i is even and 8≤i≤64}8≤i≤64}; and L(n)={n}∪{i∣iL(n)={n}∪{i∣i is even, 8≤i≤n2n8≤i≤n2n, and i≠10}i≠10} if nn is even and n≥6n≥6.