On Potentially \({}_{k}C_\ell\)-Graphic Sequences

For given integers \( k \) and \( \ell \), \( 3 \leq k \leq \ell \), a graphic sequence \( \pi = (d_1, d_2, \dots, d_n) \) is said to be potentially \({}_{k}C_\ell\)-graphic if there exists a realization of \( \pi \) containing \( C_r \), for each \( r \), where \( k \leq r \leq \ell \) and \( C_r \) is the cycle of length \( r \). Luo (Ars Combinatoria 64(2002)301-318) characterized the potentially \( C_\ell \)-graphic sequences without zero terms for \( r = 3, 4, 5 \). In this paper, we characterize the potentially \(\prescript{}{k}C_\ell\)-graphic sequences without zero terms for \( k = 3, 4 \leq \ell \leq 5 \) and \( k = 4, \ell = 5 \).