Turan number for $p{S}_r$

Abstract

The Turán number $ex(m,G)$ of the graph $G$ is the maximum number of edges of an $m$-vertex simple graph having no $G$ as a subgraph. A \emph{star} $S_r$ is the complete bipartite graph $K_{1,r}$ (or a tree with one internal vertex and $r$ leaves) and $p{S}_r$ denotes the disjoint union of $p$ copies of $S_r$. A result of Lidický et al. (Electron. J. Combin. $20(2)(2013)P62$) implies that $ex(m,p{S}_r)=\lfloor \frac{(m-p+1)(r-1)}{2}\rfloor +(p-1)m–(\genfrac{}{}{0ex}{}{p}{2})$ for $m$ sufficiently large. In this paper, we give another proof and show that $ex(m,p{S}_r)=\lfloor \frac{(m-p+1)(r-1)}{2}\rfloor +(p-1)m–(\genfrac{}{}{0ex}{}{p}{2})$ for all $r\ge 1$, $p\ge 1$, and $m\ge \frac{1}{2}{r}^{2}p(p–1)+p–2+max\{rp,r^2+2r\}$.