The Turan Number of Disjoint Copies of Paths

Abstract

The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we determine the value $ex(n, k\cdot P_3)$ and characterize all extremal graphs. This extends a result of Bushaw and Kettle [N. Bushaw and N. Kettle, Tur\'{a}n Numbers of multiple and equibipartite forests, Combin. Probab. Comput., 20(2011) 837-853.], which solved the conjecture proposed by Gorgol in [I. Gorgol. Tur\'{a}n numbers for disjoint copies of graphs. {\it Graphs Combin.}, 27 (2011) 661-667.].