On the number of 4-contractible edges in plane triangulations

TL;DR

This paper refines bounds on the minimum number of contractible edges preserving 4-connectedness in 4-connected plane triangulations, providing new lower bounds and identifying extremal graphs.

Contribution

It improves previous bounds on contractible edges in 4-connected plane triangulations and characterizes extremal cases.

Findings

At least |V_{≥5}| + 2 contractible edges exist in such graphs.

Established lower bounds for the number of contractible edges.

Identified extremal graphs achieving these bounds.

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.