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The planar Turan number of double star S_(3,5)
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Given a graph H and a positive integer n, the planar Turan number of H, denoted by exp(n, H), is the maximum number of edges in an n-vertex H-free planar graph.D.Ghosh, et al.initiated the topic of double stars S_(k,l). Recently Xu et al.[AIMS Mathematics, 2025, 10(1): 1628-1644.] mentioned that exp(n, S_(3,5)) is still unknown.In this paper, we first establish that the planar Turan number S_(3,5) satisfies exp(n, S_(3,5)) <= 23n/8 - 9/2 for all n >= 2. The upper bound is tight for n = 12.
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Planar Tur\'an number of quasi-double stars
For quasi-double stars W_{h,k} with 1≤h≤2≤k≤5, the paper proves planar Turán bounds of 3(h+k)/(h+k+2)n for h+k≤5, and two-sided bounds of 5/2 n and 17/6 n for larger cases.
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