Turán Numbers for Vertex-disjoint Triangles and Pentagons

The Turán number, denoted by 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_ℓ the union of k vertex-disjoint copies of C_ℓ. 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 ex (n, kC₃). We determine the Turán number ex(n, kC₃) for \(n \geq {k^{2}+5k \over 2}\) when k ≤ 4, and n ≥ k² + 2 when k ≥ 4. Moreover, we give lower and upper bounds for ex (n, kC₃) with \(3k \leq n \leq {k^{2}+5k \over 2}\) when k ≤ 4, and 3k ≤ n ≤ k² + 2 when k ≥ 4. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ex (n, kC₅). Finally, we determine the Turán number ex (n, kC₅) for n = 5k, and propose two conjectures for ex (n, kC₅) for the other values of n.