Tur\'{a}n number of four vertex-disjoint cliques

TL;DR

This paper determines the maximum number of edges in large graphs that do not contain four disjoint cliques of size p, extending classical extremal graph theory results.

Contribution

It explicitly calculates the Turán number for the union of four disjoint cliques of size p using advanced combinatorial techniques.

Findings

Derived exact Turán numbers for 4K_p for all n and p ≥ 3.

Applied the Hajnal-Szemerédi theorem and discharging method in the proof.

Extended classical extremal graph theory results to a new class of forbidden subgraphs.

Abstract

Given a graph $H$, the Tur\'{a}n number ${\rm ex}(n,H)$ of $H$ is the maximum number of edges of an $n$-vertex simple graph containing no $H$ as a subgraph. Let $kK_p$ denote the disjoint union of $k$ copies of the complete graph $K_p$. In this paper, utilizing the idea of the proof of the Hajnal-Szemer\'{e}di Theorem and discharging, we determine the value ${\rm ex}(n,4K_p)$ for all $n$ and $p\ge 3$.