Planar Turán Number of the Θ6

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 ex p (𝑛, F ), is the maximum number of edges in an 𝑛-vertex F -free planar graph. Let Θ 𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound ex p (𝑛, Θ 6 ) ≤ 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 ex p (𝑛, Θ 6 ) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ 6 -free planar graph attaining the bound.