Fangfang Wu, Hajo Broersma, Shenggui Zhang, Binlong Li
The Turán number, denoted by ${\rm ex}\,(n,H)$, is the maximum number of edges of a graph on $n$ vertices containing no graph $H$ as a subgraph. Denote by $kC_{\ell}$ the union of $k$ vertex-disjoint copies of $C_{\ell}$. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ${\rm ex}\,(n, kC_{3})$. We determine the Turán number ${\rm ex}(n, kC_{3})$ for $n\geq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $n\geq k^{2}+2$ when $k\geq4$. Moreover, we give lower and upper bounds for ${\rm ex}\,(n, kC_{3})$ with $3k\leq n\leq\frac{k^{2}+5k}{2}$ when $k\leq4$, and $3k\leq n\leq k^{2}+2$ when $k\geq4$. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ${\rm ex}\,(n, kC_{5})$. Finally, we determine the Turán number ${\rm ex}\,(n, kC_{5})$ for $n=5k$, and propose two conjectures for ${\rm ex}\,(n, kC_{5})$ for the other values of $n$.
