Generalized Zykov's Theorem
Rajat Adak, L. Sunil Chandran

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 , let denote its number of vertices, and let denote the number of copies of in . Zykov's theorem (1949) asserts that for any -free graph and , \[ N(G,K_t) \le {r \choose t}\left(\frac{n}{r}\right)^t \] We generalize Zykov's bound within a vertex-based localization framework. For each vertex , let denote the order of the largest clique containing . 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 is a regular complete multipartite graph. \newline Note that if we impose the condition that, is -free, then for all . Thus, plugging for all , we retrieve Zykov's bound.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
