Tur\'an number of complete bipartite graphs with bounded matching number

TL;DR

This paper determines the exact maximum number of edges in large graphs that avoid certain complete bipartite subgraphs with bounded matching number, extending previous results to more complex bipartite structures.

Contribution

It provides exact Turán numbers for graphs avoiding specific complete bipartite graphs with bounded matching number, generalizing earlier findings.

Findings

Exact Turán number for $ex(n,igl ext{K}_{l,t}, M_{s+1}igr)$ when $s, n$ are large.

Explicit formula for $ex(n, ext{K}_{2,2}, M_{s+1})$ for large $s$.

Explicit formula for $ex(n, ext{K}_{2,t}, M_{s+1})$ when $t extgreater 3$ and $s$ large.

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 Tur\'an number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $M_{s}$ be the matching consisting of $ s $ independent edges. Recently, Alon and Frank determined the exact value of $ex(n,\{K_{m},M_{s+1}\})$. Gerbner obtained several results about $ex(n,\{F,M_{s+1}\})$ when $F$ satisfies certain proportions. In this paper, we determine the exact value of $ex(n,\{K_{l,t},M_{s+1}\})$ when $s, n$ are large enough for every $3\leq l\leq t$. When $n$ is large enough, we also show that $ex(n,\{K_{2,2}, M_{s+1}\})=n+{s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ for $s\ge 12$ and $ex(n,\{K_{2,t},M_{s+1}\})=n+(t-1){s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ when $t\ge 3$ and $s$ is large enough.