Turán number of four vertex-disjoint cliques

TL;DR

The paper resolves the Turán number ${\rm ex}(n,4K_p)$ for all integers $n$ and $p\ge3$ by combining Hajnal–Szemerédi equitable coloring ideas with discharging methods. A key component is Theorem 2.7, which characterizes near-extremal graphs on $n=4p-1+s$ vertices with $|E(G)|\le7s$ and maximum degree $\Delta(G)\le6$, showing the graph must be of the form $(s/3)K_7\cup\overline{K}_{n-7s/3}$. Using this structural result, the authors derive exact values for small $n$ (explicit in Theorem 1.8) and a complete large-$n$ formula, ${\rm ex}(n,4K_p)=3+3(n-1)+t_{n-3,p-1}$ for $n\ge7p-1$, with extremal graph $K_3\vee T_{n-3,p-1}$. The results advance the understanding of Turán numbers for multiple disjoint copies of cliques and lay groundwork for potential generalizations to longer families of forbidden subgraphs.

Abstract

Given a graph $H$, the Turán 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-Szemerédi Theorem and discharging, we determine the value ${\rm ex}(n,4K_p)$ for all $n$ and $p\ge 3$.