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

摘要: Summary: "In this paper we consider a variation of the classical Turán-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 $\sigma(S)\geq l$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 $m\geq 3$, $n\geq 3m$; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$, for $m\geq 3$, $n\geq 5m-2$.''