On Potentially ${}_k{C}_{\ell }$-Graphic Sequences

Abstract

For given integers $k$ and $\ell$, $3\le k\le \ell$, 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 $k\le r\le \ell$ 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 $\text{prescript}k{C}_{\ell }$-graphic sequences without zero terms for $k=3,4\le \ell \le 5$ and $k=4,\ell =5$.