The Turán number of spanning star forests

Abstract

Let F be a family of graphs. The Turán number of F , denoted by ex(n,F), is the maximum number of edges in a graph with n vertices which does not contain any subgraph isomorphic to some graph in F . A star forest is a forest whose connected components are all stars and isolated vertices. Motivated by the results of Wang, Yang and Ning about the spanning Turán number of linear forests [J. Wang and W. Yang, The Turán number for spanning linear forests, Discrete Appl. Math. 254 (2019) 291–294; B. Ning and J. Wang, The formula for Turán number of spanning linear forests, Discrete Math. 343 (2020) 111924]. In this paper, let Sn,k be the set of all star forests with n vertices and k edges. We prove that when 1 ≤ k ≤ n− 1, ex(n,Sn,k) = ⌊ k−1 2 ⌋ .