Variations of Two Classical Turdn-Type Extremal Results

Abstract

We consider a variation of a classical Turán-type extremal problem due to Bollobás $[2,p.398,no.13]$ as follows: determine the smallest even integer $\sigma (C^k,n)$ such that every graphic sequence $\pi =(d_1,d_2,\dots ,d_n)$ with term sum $\sigma (\pi )=d_1+d_2+\cdots +d_n\ge \sigma (C^k,n)$ has a realization $G$ containing a cycle with $k$ chords incident to a vertex on the cycle. Moreover, we also consider a variation of a classical Turán-type extremal result due to Faudree and Schelp $[7]$ as follows: determine the smallest even integer $\sigma (P_{\ell },n)$ such that every graphic sequence $\pi =(d_1,d_2,\dots ,d_n)$ with $\sigma (\pi )\ge \sigma (P_{\ell },n)$ has a realization $G$ containing $P_{\ell }$ as a subgraph, where $P_{\ell }$ is the path of length 2. In this paper, we determine the values of $\sigma (P_{\ell },n)$ for $n\ge \ell +1$ and the values of $\sigma (C^k,n)$ for $n\ge (k+3)(2k+5)$.