The maximum number of cliques in disjoint copies of graphs

TL;DR

The paper addresses the problem of maximizing the number of $s$-cliques in $n$-vertex graphs that avoid $k$ pairwise disjoint copies of a fixed graph $H$, i.e., ex$(n, K_s, kH)$. It develops both lower-bound constructions and upper-bound arguments, yielding general bounds and exact values in several regimes, including $H$ a path-graph and small $k$ ($k=2,3$). A central finding is that, for many cases, the extremal graph is the join $K_{k-1} + M_{n-k+1}$, where $M_{n-k+1}$ is a matching/independent-vertex structure, illustrating a recurring extremal structure in disjoint-union Turán problems. The results confirm conjectures by Chen, Yang, Yuan, and Zhang in key scenarios and advance understanding of generalized Turán numbers for disjoint unions, with implications for cliques in $H$-free graphs and related extremal combinatorics questions.

Abstract

The problem of determining the maximum number of copies of $T$ in an $H$-free graph, for any graphs $T$ and $H$, was considered by Alon and Shikhelman. This is a variant of Turán's classical extremal problem. We show lower and upper bounds for the maximum number of $s$-cliques in a graph with no disjoint copies of arbitrary graph. We also determine the maximum number of $s$-cliques in an $n$-vertex graph that does not contain a disjoint union of $k$ paths of length two when $k=2,3$, or $s\geqslant k+2$, or $n$ is sufficiently large, this partly confirms a conjecture posed by Chen, Yang, Yuan, and Zhang \cite{2024Chen113974}.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 3.1: 2018Luo219
• Lemma 4.1
• Proposition 5.1