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- Title
A Note on the Turán Number of an Arbitrary Star Forest.
- Authors
Wang, Bing; Yin, Jian-Hua
- Abstract
The Turán number of a graph G, denoted by ex(n, G), is the maximum number of edges of an n-vertex simple graph having no G as a subgraph. Let S ℓ denote the star with ℓ + 1 vertices, and let ⋃ i = 1 k S ℓ i denote the disjoint union of S ℓ 1 , … , S ℓ k . For k ≥ 2 and ℓ 1 ≥ ⋯ ≥ ℓ k ≥ 1 , Lidický et al. [On the Turán number of forests, Electron. J. Combin., 20(2)(2013)#P62] proved that e x (n , ⋃ i = 1 k S ℓ i) = max 1 ≤ i ≤ k { ⌊ (ℓ i + 2 i - 3) n - (i - 1) (ℓ i + i - 1) 2 ⌋ } for n sufficiently large. In this paper, we further show that e x (n , ⋃ i = 1 k S ℓ i) = max 1 ≤ i ≤ k (ℓ i + 2 i - 3) n - (i - 1) (ℓ i + i - 1) 2 <graphic href="373_2022_2502_Article_Equ1.gif"></graphic> for n ≥ N k by giving another proof, where L j = max 1 ≤ i ≤ j { ℓ i + 2 i - 3 } for 2 ≤ j ≤ k and N k = max 2 ≤ j ≤ k (∑ i = 1 j - 1 ℓ i + 1) (∑ i = 1 j 2 ℓ i - ℓ j + 2 j - 5) + (j - 1) ℓ 2 - (j - 2) ℓ j + j 2 - 3 j + 5 L j - ℓ j - 2 j + 5 .
- Publication
Graphs & Combinatorics, 2022, Vol 38, Issue 3, p1
- ISSN
0911-0119
- Publication type
Article
- DOI
10.1007/s00373-022-02502-1