Degree Powers in $$C_5$$ C 5 -Free Graphs

Abstract

Let \(G\) be a graph with degree sequence \(d_1,d_2,\ldots ,d_n\). Given a positive integer \(p\), denote by \(e_p(G)=\sum _{i=1}^n d_i^p\). Caro and Yuster introduced a Turan-type problem for \(e_p(G)\): given an integer \(p\), how large can \(e_p(G)\) be if \(G\) has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order \(r+1\) and a cycle of even length, respectively. Denote by \(ex_p(n,H)\) the maximum value of \(e_p(G)\) taken over all graphs with \(n\) vertices that do not contain \(H\) as a subgraph. Clearly, \(ex_1(n,H)=2ex(n,H)\), where \(ex(n,H)\) denotes the classical Turan number. In this paper, we consider \(ex_p(n, C_5)\) and prove that for any positive integer \(p\) and sufficiently large \(n\), there exists a constant \(c=c(p)\) such that the following holds: if \(ex_p(n, C_5)=e_p(G)\) for some \(C_5\)-free graph \(G\) of order \(n\), then \(G\) is a complete bipartite graph having one vertex class of size \(cn+o(n)\) and the other \((1-c)n+o(n)\).