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 \(\pi = (d_1,d_2,\ldots,d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(C^k,n)\) 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 \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) \geq \sigma(P_\ell,n)\) 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 \(n \geq \ell+1\) and the values of \(\sigma(C^k,n)\) for \(n \geq (k+3)(2k+5)\).