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On the number of 4-contractible edges in plane triangulations

Toshiki Abe, Michitaka Furuya, Raiji Mukae, Shoichi Tsuchiya

Abstract

In 2007, Ando and Egawa proved a theorem which provides a lower bound on the number of contractible edges preserving $4$-connectedness in $4$-connected graphs. In this paper, we refine their bounds, especially for the $4$-connected plane triangulations. In particular, we show that if $G$ is a $4$-connected plane triangulation of order at least $7$, then $G$ contains at least $|V_{\ge 5}|+2$ contractible edges preserving $4$-connectedness, where $V_{\ge 5}$ is the set of vertices of degree at least $5$. We also determine the extremal graphs.

On the number of 4-contractible edges in plane triangulations

Abstract

In 2007, Ando and Egawa proved a theorem which provides a lower bound on the number of contractible edges preserving -connectedness in -connected graphs. In this paper, we refine their bounds, especially for the -connected plane triangulations. In particular, we show that if is a -connected plane triangulation of order at least , then contains at least contractible edges preserving -connectedness, where is the set of vertices of degree at least . We also determine the extremal graphs.

Paper Structure

This paper contains 2 sections, 5 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem A

Let $G$ be a $4$-connected graph. Then we have $|E_4| \ge \frac{1}{34} \cdot (|E(G)|-2|G|)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The plane graph $G_0$ (white vertices indicate $V_4$ and black vertices indicate $V_{\ge 5}$).
  • Figure 2: Two transformations (white vertices indicate $V_4$ and black vertices indicate $V_{\ge 5}$ ).

Theorems & Definitions (17)

  • Theorem A: Ando, Egawa, Kawarabayashi and Kriesell AEKK
  • Theorem B: Ando and Egawa AE
  • Theorem C: McCuaig, Haglin and Venkatesan MHV
  • Theorem 1
  • Lemma 2
  • Claim 2.1
  • Claim 2.2
  • Claim 2.3
  • Claim 2.4
  • Claim 2.5
  • ...and 7 more