Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dense $2$-connected planar graphs and the planar Turán number of $2C_k$

Ping Li

Abstract

Shi, Walsh and Yu demonstrated that any dense planar graph with certain property (known as circuit graph) contains a large near-triangulation. We extend the result to $2$-connected plane graphs, thereby addressing a question posed by them. Using the result, we prove that the planar Tuán number of $2C_k$ is $\left[3-Θ(k^{\log_23})^{-1}\right]n$ when $k\geq 5$.

Dense $2$-connected planar graphs and the planar Turán number of $2C_k$

Abstract

Shi, Walsh and Yu demonstrated that any dense planar graph with certain property (known as circuit graph) contains a large near-triangulation. We extend the result to -connected plane graphs, thereby addressing a question posed by them. Using the result, we prove that the planar Tuán number of is when .

Paper Structure

This paper contains 6 sections, 14 theorems, 46 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm-near']}
  4. Proof of Theorem \ref{['thm-2ck']}
  5. Concluding Remark
  6. Acknowledgements

Key Result

Theorem 1.1

For all $k\geq 3$, if $(G,C)$ is a circuit graph with at least $k^{\log_23}$ vertices, then $G$ has a cycle of length at least $k$.

Figures (5)

  • Figure 1: An example.
  • Figure 2: The plane subgraph of $G$ bounded by $C^1$.
  • Figure 3: Peripheral neighborhoods of $v$.
  • Figure 4: Subgraphs $D_1$ and $D_2$ (left side) Three paths $P_1,L_1$ and $L_2$ (right side).
  • Figure 5: Some notations on $B_s$.

Theorems & Definitions (27)

  • Theorem 1.1: Chen and Yu CY
  • Theorem 1.2: Shi, Walsh and Yu SWY
  • Theorem 1.3: Shi, Walsh and Yu SWY
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 17 more