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October 15, 2025Open Access

Planar Turán number of quasi-double stars

Key Points

The study finds that ex_P(n,W_{h,k}) is maximized under specific conditions, especially for h+k values.
Tight bounds were established, with ex_P(n,W_{h,k}) ≤ (3(h+k)/(h+k+2))n when 3 ≤ h+k ≤ 5.
Results show exact equality for conditions like (h+k+2)|n, highlighting specific integer relationships.
Inequalities for ex_P(n,W_{2,4}) and ex_P(n,W_{2,5}) give further insights into edge limits for these structures.

Abstract

Given a graph H, we call a graph H-free if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by ex₏ (n, H), is the maximum number of edges in a planar H-free graph on n vertices. A (h, k) -quasi-double star W₇, ₊, obtained from a path P₃=v₁v₂v₃ by adding h leaves and k leaves to the vertices v₁ and v₃, respectively, is a subclass of caterpillars. In this paper, we study ex₏ (n, W₇, ₊) for all 1 h 2 k 5, and obtain some tight bounds ex₏ (n, W₇, ₊) 3 (h+k) h+k+2n for 3 h+k 5 with equality holds if (h+k+2) n, and ex₏ (n, W₁, ₅) 52n with equality holds if 12 n. Also we show that 94n ex₏ (n, W₂, ₄) 52n and 52n ex₏ (n, W₂, ₅) 176n, respectively.

Planar Turán number of quasi-double stars

Key Points

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