The shifting method and generalized Tur\'{a}n number of matchings

TL;DR

This paper determines the maximum number of certain subgraphs in graphs avoiding specific matchings, using the shifting method, and extends results to bipartite graphs with explicit formulas.

Contribution

It provides exact formulas for generalized Turán numbers involving matchings and complete bipartite graphs, employing the shifting method for proofs.

Findings

Exact formulas for $ex(n,K_s,M_{k+1})$ and $ex(n,K_{s,t}^*,M_{k+1})$

Explicit bounds for bipartite case $ex_{bip}(n,K_{s,t},M_{k+1})$

Application of the shifting method to Turán-type problems

Abstract

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\'{a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\'{a}n number $ex(n,F)$. Let $M_{k}$ be a matching with $k$ edges and $K^{*}_{s,t}$ a graph obtained from $K_{s,t}$ by replacing the part of size $s$ by a clique of the same size. In this paper, we show that for any $s\geq 2$ and $n\geq 2k+1$, \[ ex(n,K_s,M_{k+1})=\max\left\{\binom{2k+1}{s}, \binom{k}{s}+(n-k)\binom{k}{s-1}\right\}. \] For any $s\geq 1$, $t\geq 2$ and $n\geq 2k+1$, \[ ex(n,K_{s,t}^*,M_{k+1})=\max\left\{\binom{2k+1}{s+t}\binom{s+t}{t}, \binom{k}{s}\binom{n-s}{t}+(n-k)\binom{k}{s+t-1}\binom{s+t-1}{t}\right\}. \] Moreover, we also study the bipartite case of the problem. Let $ex_{bip}(n,T,F)$ be the maximum possible number of copies of $T$ in an $F$-free bipartite graph with each part of size $n$. We prove that for any $s,t\geq 1$ and $n\geq k$, \[ ex_{bip}(n,K_{s,t},M_{k+1})=\left\{ \begin{aligned} &\binom{k}{s}\binom{n}{t}+\binom{k}{t}\binom{n}{s}, & \quad s\neq t, &\binom{k}{s}\binom{n}{s},&\quad s=t. \end{aligned} \right. \] Our proof is mainly based on the shifting method.