The Turán Numbers of Special Forests

A graph is called H-free if it does not contain H as a subgraph. The Turán number of H, denoted by ex(n, H), is the maximum number of edges in any H-free graph on n vertices. For sufficiently large n, Lidický et al. (Electron J Combin 20:62, 2013) determined ex(n, F), where F denotes a class of forest with components each of order 4, and they characterized all the extremal graphs. Moreover, Lan et al. (Appl Math Comput 348: 270–274, 2019) proved that the extremal graph is unique when n is large enough. Motivated by their results, we consider a class of forests F with each component a path or star of order 6, we determine ex(n, F) for sufficiently large n and we also characterize all the extremal graphs. Furthermore, we prove that the extremal graph is unique when n is large enough. Let kP_n denote the disjoint union of k copies of the path P_n on n vertices. Recently, Lan et al. (Discuss Math Graph Theory 39: 805–814, 2019) determined the exact value of ex(n, 2 P₇). Motivated by their result, we show that ex(n, 2 P₉) = max{ (7 n+ 153 + r(r- 8)) / 2 , 7 n- 27 } for all n≥ 18 , where r is the remainder of n- 17 when divided by 8.