Generalized Zykov's Theorem

TL;DR

This paper extends Zykov's theorem by providing a vertex-based bound on the number of cliques in a graph, which generalizes the classical result and characterizes extremal graphs.

Contribution

It introduces a localized bound involving the largest clique size per vertex, generalizing Zykov's theorem and characterizing equality cases as regular complete multipartite graphs.

Findings

Derived a new upper bound for clique counts based on vertex clique sizes

Showed that the bound reduces to Zykov's classical result under certain conditions

Characterized the extremal graphs where equality holds

Abstract

For a simple graph $G$, let $n$ denote its number of vertices, and let $N(G,K_t)$ denote the number of copies of $K_t$ in $G$. Zykov's theorem (1949) asserts that for any $K_{r+1}$-free graph and $t \ge 2$, \[ N(G,K_t) \le {r \choose t}\left(\frac{n}{r}\right)^t \] We generalize Zykov's bound within a vertex-based localization framework. For each vertex $v \in V(G)$, let $c(v)$ denote the order of the largest clique containing $v$. In this paper, we show that \[ N(G,K_t) \le n^{t-1} \sum_{v \in V(G)} \frac{1}{c(v)^t} {c(v) \choose t} \] We further show that equality holds if and only if $G$ is a regular complete multipartite graph. \newline Note that if we impose the condition that, $G$ is $K_{r+1}$-free, then $c(v) \leq r$ for all $v \in V(G)$. Thus, plugging $c(v) = r$ for all $v \in V(G)$, we retrieve Zykov's bound.