The planar Tur\'an number of {K4,C5}\{K_4,C_5\} and {K4,C6}\{K_4,C_6\}

Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.Comment: 11 pages, 11 figures. arXiv admin note: text overlap with arXiv:2307.0690

Dense circuit graphs and the planar Tur\'an number of a cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Tur\'an numbers. In particular, we prove that there is a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 - Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and Shantanam

The planar Tur\'an number of the seven-cycle

The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$ and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over 7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph

Planar Tur\'an Number of the Θ6\Theta_6

Let $F$ be a graph. The planar Tur\'an number of $F$, denoted by $\text{ex}_{\mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex planar graph containing no copy of $F$ as a subgraph. Let $\Theta_k$ denote the family of Theta graphs on $k\geq 4$ vertices, that is, a graph obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Y. Lan, et.al. determined sharp upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_4)$ and $\text{ex}_{\mathcal{P}}(n,\Theta_5)$. Moreover, they obtained an upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_6)$. They proved that, $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$. In this paper, we improve their result by giving a bound which is sharp. In particular, we prove that $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{48}{7}$ and demonstrate that there are infinitely many $n$ for which there exists a $\Theta_6$-free planar graph $G$ on $n$ vertices, which attains the bound.Comment: 23 pages, 19 figure

Planar Tur\'an number of the 7-cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for $\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} - \frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely many integers $n$

Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem fully rewritten, fixes a serious error in the previous version found by Kevin Hendre