Vertex-Based Localization of Turán's Theorem
Rajat Adak, L. Sunil Chandran
TL;DR
The paper introduces a vertex-based localization of Turán's theorem by assigning to each vertex v a weight c(v) = the largest clique containing v and proving m ≤ ⌊(n/2)∑_{v} (c(v)−1)/c(v)⌋. It completely characterizes equality cases, showing that extremal graphs are precisely the Turán graphs in a defined class 𝒮 together with two exceptional non-Turán graphs X (paraglider) and Y, clarifying when the standard floor-bound is tight. The work also demonstrates how the classic floor-form bound can be recovered from the vertex-based result and discusses a vertex-based localization for the weighted Turán bound (wturan), including a non-floor variant that yields the regular Turán graphs as extremal. Overall, it provides a precise, local-structure-driven framework for extremal Turán-type problems and clarifies the landscape of extremal graphs beyond the standard Turán family.
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 $m \leq |E(T(n, r))|,$ 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, $m \leq \left\lfloor \frac{n^2(r-1)}{2r} \right\rfloor.$ 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 $\mathcal{S}$.