Turán number of complete bipartite graphs with bounded matching number

Abstract

Let $F$ be a family of graphs. A graph G is $F$-free if G does not contain any $F\in F$ as a subgraph. The Turán number $\text{ex}(n,F)$ is the maximum number of edges in an n-vertex $F$-free graph. Let $M_s$ be the matching consisting of s independent edges. Recently, Alon and Frankl determined the exact value of $\text{ex}(n,\{K_m,M_{s+1}\left\}\right)$. Gerbner obtained several results about $\text{ex}(n,\{F,M_{s+1}\left\}\right)$ when F satisfies certain properties. In this paper, we determine the exact value of $\text{ex}(n,\{K_{\ell ,t},M_{s+1}\left\}\right)$ when $s,n$ are large enough for every $3\le \ell \le t$. When n is large enough, we also show that $\text{ex}(n,\{K_{2,2},M_{s+1}\left\}\right)=n+\left(\begin{array}{c}s\\ 2\end{array}\right)-\lceil \frac{s}{2}\rceil$ for $s\ge 12$ and $\text{ex}(n,\{K_{2,t},M_{s+1}\left\}\right)=n+(t-1)\left(\begin{array}{c}s\\ 2\end{array}\right)-\lceil \frac{s}{2}\rceil$ when $t\ge 3$ and s is large enough.