Cycles in cube-connected cycles graphs

Abstract

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