Planar Turán number of two adjacent cycles

TL;DR

The paper resolves the exact planar Turán numbers for two adjacent cycles by introducing and exploiting 3-face-block analyses in plane graphs. The authors develop a framework around $3$-face-blocks and the partition $R_v$ to derive tight upper bounds, then construct explicit extremal graphs that achieve these bounds for all relevant $n$, yielding piecewise formulas: $ex_{\mathcal{P}}(n, C_3\text{-}C_3)=3n-6$ for $n\le5$, $3n-7$ for $n=6$, and $ex_{\mathcal{P}}(n, C_3\text{-}C_3)=\lceil \frac{5n}{2} \rceil -5$ for $n\ge7$; and $ex_{\mathcal{P}}(n, C_3\text{-}C_4)=3n-6$ for $n\le6$, $3n-7$ for $n=7$, and $ex_{\mathcal{P}}(n, C_3\text{-}C_4)=\lfloor \frac{5n}{2} \rfloor -4$ for $n\ge8$. The results also yield corollaries for $2C_3$ and $C_3\text{-}C_4$, including a correction to a known value at $n=6$. Overall, the work advances the precise characterization of planar extremal graphs for near-cycle structures and demonstrates a robust method for analyzing forbidden subgraphs in planar settings.

Abstract

The planar Turán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k(k\geq 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Turán number of $C_{3}\text{-}C_{3}$ and $C_{3}\text{-}C_{4}$, where $C_{k}\text{-}C_{\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them.

Figures (7)

• Figure 1: $R_v$ with the partition $(2,2,1,1)$.
• Figure 2: The $3$-face-blocks with $\sum_{v \in B}|\overline{R_v}|\geq3|V(B)|$
• Figure 3: The extremal graph with $n=11$.
• Figure 4: The $3$-face-blocks with $\sum_{v \in B}|\overline{R_v}| > 3|V(B)|$.
• Figure 5: Relabel the vertices.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Corollary 1.2
• Definition 2.1
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof