Planar Turán Number of the Θ6

Chuanqi, Xiao and Ghosh, Debarun and Győri, Ervin and Addisu, Paulos and Zamora, Oscar (2024) Planar Turán Number of the Θ6. STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA, 61 (2). pp. 89-115. ISSN 0081-6906

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Abstract

Let F be a nonempty family of graphs. A graph G is called F -free if it contains no graph from F as a subgraph. For a positive integer n, the planar Turán number of F, denoted by exP (n, F), is the maximum number of edges in an n-vertex F -free planar graph. Let Θk be the family of Theta graphs on k ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive vertices of a k-cycle with an edge. Lan, Shi and Song determined an upper bound exP (n, Θ6) ≤ 18n/7−36n/7, but for large n, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exP (n, Θ6) ≤ 18n/−48n/7 and then we demonstrate the existence of infinitely many positive integer n and an n-vertex Θ6-free planar graph attaining the bound. © 2024 Akadémiai Kiadó, Budapest.

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