The Number of Copies of $K_{2,t+1}$ in a Graph

Abstract

It is not yet finished to locate the exact value of ex$(n;K_{2,t+1})$ in the extremal graph theory. Füredi (J Combin Theory Ser A 75(1):141–144, 1996) proved that the Hyltén-Cavallius’ bound $\frac{n}{4}\left( \sqrt{4tn-4t+1}+1 \right)$ is asymptotically best possible for ex$(n;K_{2,t+1})$. In this paper we investigate the number of copies of $K_{2,t+1}$ ($t\ge 2$) in a graph. We show that an n-vertex graph G contains at least $\frac{\epsilon }{t+1}\sqrt{4tn-4t+1}+\frac{8t\epsilon ^2}{(t+1)(n-1)}+\frac{2\epsilon ^2}{(t+1)n}$ copies of $K_{2,t+1}$ provided that $|E(G)|=\frac{n}{4}\left( \sqrt{4tn-4t+1}+1 \right) +\epsilon$, where $\epsilon >0$ is a real number. Furthermore, we also obtain a formula for the number of copies of $K_{2,t+1}$ in a graph.