Two results about the Turán number of star forests

Abstract

The Turán number of a graph H, denoted by ex(n,H), is the maximum number of edges of an n-vertex simple graph having no H as a subgraph. Let Sℓ denote the star on ℓ+1 vertices, and let k⋅Sℓ denote k disjoint copies of Sℓ. Erdős and Gallai determined the value ex(n,k⋅S1) for all positive integers k and n. Yuan and Zhang determined the value ex(n,k⋅S2) and characterized all extremal graphs for all positive integers k and n. Recently, Lan et al. determined the value ex(n,2⋅S3) for all positive integers n. In this paper, we first determine the Turán number ex(n,2⋅Sℓ) for all positive integers ℓ(≥4) and n, and then determine the Turán number ex(n,3⋅Sℓ) for all positive integers ℓ(≥3) and n, improving two of the results of Lidický et al.