An Important Property about Vertex Distance of Crossed Cubes CQn

The crossed cube is an important variant of the hypercube Q_n, it has many properties superior to hypercube. In this paper, we studied an important property about vertex distance in CQ_n, that is, for any vertex u_L ∈ L, v_L is a neighbor of u_L in L, and u_R, v_R be neighbors of u_L, u_L in R, respectively, then, the distance between u_R and v_R is 1 or 2 or 3. According to the result, we get a corollary that for any u in CQ_n and any integer 8 ≤ ℓ ≤ 2ⁿ, there exists cycle C of length ℓ such that u is on C if the number of faulty edges |F| does not exceed n - 3.