An improved lower bound for the planar Turán number of cycles

The planar Turán number of a graph $H$, denoted by $e{x}_{{}_{\mathcal{P}}}(n,H)$, is the largest number of edges in a planar graph on $n$ vertices without containing $H$ as a subgraph. In this paper, we continue to study the topic of “extremal” planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213–230]. We first obtain an improved lower bound for $e{x}_{{}_{\mathcal{P}}}(n,C_k)$ for all $k\ge 13$ and $n\ge 5(k-6+\lfloor (k-1)\mathrm{/}2\rfloor )(k-1)\mathrm{/}2$; the construction for each $k$ and $n$ provides a simpler counterexample to a conjecture of Ghosh, Győri, Martin, Paulos and Xiao [arxiv:2004.14094v1], which has recently been disproved by Cranston, Lidický, Liu and Shantanam [Electron. J. Combin. 29(3) (2022) #P3.31] for every $k\ge 11$ and $n$ sufficiently large (as a function of $k$). We then prove that $e{x}_{{}_{\mathcal{P}}}(n,H^{+})=e{x}_{{}_{\mathcal{P}}}(n,H)$ for all $k\ge 5$ and $n\ge \mathrm{\mid }H\mathrm{\mid }+1$, where $H\in \{C_k,2C_k\}$ and $H^{+}$ is obtained from $H$ by adding a pendant edge to a vertex of degree two.