Generalized Tur\'an problems for a matching and long cycles

TL;DR

This paper determines the maximum number of copies of a complete graph in large graphs that avoid long cycles and matchings, extending classical Turán problems with exact results and extremal structures.

Contribution

It extends Turán-type results to include long cycles and matchings, providing exact values and extremal graphs for large graphs.

Findings

Exact value of $ex(n,K_r, ext{for large } n)$ for graphs avoiding long cycles and matchings.

Characterization of extremal graphs when the cycle length parameter is odd.

Precise extremal numbers and structures for avoiding long cycles and matchings.

Abstract

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The general Tur\'an number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$ in an $n$-vertex $\mathscr{F}$-free graph. Then $ex(n, K_2,\mathscr{F})$, also denote by $ex(n, \mathscr{F})$, is the Tur\'an number. Recently, Alon and Frankl determined the exact value of $ex(n, \{K_{k},M_{s+1}\})$, where $K_{k}$ and $M_{s+1}$ are a complete graph on $k $ vertices and a matching of size $s +1$, respectively. Then many results were obtained by extending $K_{k}$ to a general fixed graph or family of graphs. Let $C_k$ be a cycle of order $k$. Denote $C_{\ge k}=\{C_k,C_{k+1},\ldots\}$. In this paper, we determine the value of $ex(n,K_r, \{C_{\ge k},M_{s+1}\})$ for large enough $n$ and obtain the extremal graphs when $k$ is odd. Particularly, the exact value of $ex(n, \{C_{\ge k},M_{s+1}\})$ and the extremal graph are given for large enough $n$.