Planar Turán number of disjoint union of $C_3$ and $C_5$

Abstract

The planar Turán number of $H$, denoted by $e{x}_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k\ge 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\ell }$ denote the cycle of length $\ell$ and $C_{\ell }\cup C_t$ denote the union of disjoint cycles $C_{\ell }$ and $C_t$. The planar Turán number $e{x}_{\mathcal{P}}(n,H)$ is known if $H=C_{\ell }\cup C_k$, where $\ell ,k\in \{3,4\}$. In this paper, we determine the value $e{x}_{\mathcal{P}}(n,C_3\cup C_5)=\lfloor \frac{8n-13}{3}\rfloor$ and characterize the extremal graphs when $n$ is sufficiently large.

11 pages, 9 figures