The smallest degree sum that yields potentially $C_k$-graphical sequence

In this paper we consider a variation of the classical Tur'{a}n-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $\sigma (S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $\sigma (S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma (H,n)$. We show $\sigma (C_{2m+1},n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $\sigma (C_{2m+2},n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$.