Generalized Zykov's Theorem

TL;DR

This work generalizes Zykov’s bound by introducing a vertex-based localization where each vertex v is labeled by c(v), the size of the largest clique containing v. The authors prove that for any t≥2, N(G,K_t) ≤ $n^{t-1}\sum_{v\in V(G)}\frac{1}{c(v)^t}{c(v)\choose t}$, with equality exactly when G is a regular complete multipartite graph; substituting c(v)≤r recovers Zykov’s bound for K_{r+1}-free graphs. The proof employs a vertex-weighted optimization on the simplex, defining Φ(G,x)=A(G,x)−B(G,x) and using transfer operations to deduce structural properties and apply Maclaurin-type inequalities. The results provide an intuitive, local framework for Zykov-type bounds and extend the localization paradigm to vertex weights, yielding a tight extremal characterization. Overall, the paper deepens the connection between local clique structure and global clique-count bounds in extremal graph theory.

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.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Definition 1
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Definition 2
• Definition 3
• Definition 4