In this paper, we characterize the potentially \(C_k\)-graphic sequence for \(k = 3, 4, 5\). These characterizations imply several theorems due to P. Erdős, M. S. Jacobson, and J. Lehel [1], R. J. Gould, M. S. Jacobson, and J. Lehel [2], and C. H. Lai [5] and [6], respectively.