Vertex-Based Localization of Turán's Theorem

Abstract

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Turán's theorem, if $G$ is $K_{r+1}$-free, then $m\le |E(T(n,r))|,$ where $T(n,r)$ denotes the Turán graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Turán's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m\le \lfloor \frac{n^2(r-1)}{2r}\rfloor .$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the class of extremal graphs for this bound, say $\mathcal{S}$, is a proper subset of the set of Turán graphs. In this paper, we generalize this result as follows: For each $v\in V(G)$, let $c(v)$ be the order of the largest clique that contains $v$. We show that \[ m \leq \left\lfloor\frac{n}{2}\sum_{v\in V(G)}\frac{c(v)-1}{c(v)}\right\rfloor\] Furthermore, we characterize the class of extremal graphs that attain equality in this bound. Interestingly, this class contains two extra non-Turán graphs other than the graphs in $\mathcal{S}$.