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September 23, 2025Open Access

Planar Turán number of disjoint union of C₃ and C₅

Puntos clave

The planar Turán number for C3 and C5 is calculated as ⌊(8n - 13)/3⌋, applicable to large n.
Characterization of extremal graphs is provided for the graph union of C3 and C5 in planar settings.
With k ≥ 3 vertex-disjoint cycles, the planar Turán number simplifies to a common value of 3n - 6.
The findings extend known results for planar Turán numbers involving smaller cycles.

Resumen

The planar Turán number of H, denoted by ex₏ (n, H), is the maximum number of edges in an n-vertex H-free planar graph. The planar Turán number of k 3 vertex-disjoint union of cycles is the trivial value 3n-6. Let C_ denote the cycle of length and C_ Cₜ denote the union of disjoint cycles C_ and Cₜ. The planar Turán number ex₏ (n, H) is known if H=C_ Cₖ, where, k \3, 4\. In this paper, we determine the value ex₏ (n, C₃ C₅) =8n-133 and characterize the extremal graphs when n is sufficiently large.

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