Maximum Independent Set of Cliques and The Generalized Mantel's Theorem

TL;DR

This paper extends classical graph theory results by introducing the concept of clique value and the clique handshaking lemma to analyze the maximum size of graphs free of certain cliques, generalizing Mantel's theorem.

Contribution

It introduces the clique value and clique handshaking lemma, providing a generalized framework to extend Mantel's theorem to larger clique-free graphs.

Findings

Derived a generalized Mantel's theorem for $K_{oldsymbol{ heta+1}}$-free graphs.

Extended the handshaking lemma to clique structures.

Provided bounds on the maximum edges in clique-free graphs.

Abstract

A complete subgraph of any simple graph $G$ on $k$ vertices is called a $k$-\emph{clique} of $G$. In this paper, we first introduce the concept of the value of a $k$-clique ($k>1$) as an extension of the idea of the degree of a given vertex. Then, we obtain the generalized version of handshaking lemma which we call it clique handshaking lemma. The well-known classical result of Mantel states that the maximum number of edges in the class of triangle-free graphs with $n$ vertices is equal to $\frac{n^{2}}{4}$. Our main goal here is to find an extension of the above result for the class of $K_{\omega+1}$-free graphs, using the ideas of the value of cliques and the clique handshaking lemma.