# Properties of minimal charts and their applications V: charts of type $(3,2,2)$

## TL;DR

This paper proves that minimal charts of type (3,2,2), characterized by specific intersection properties of their labels and vertices, do not exist, advancing the understanding of chart properties in topological studies.

## Contribution

It establishes the non-existence of minimal charts of type (3,2,2), providing new insights into the structure of such charts in topological graph theory.

## Key findings

- No minimal chart of type (3,2,2) exists.
- The intersection properties constrain chart configurations.
- Results contribute to classification of minimal charts.

## Abstract

Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(3,2,2)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=3$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=2$, and $w(\Gamma_{m+2}\cap\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$.