Properties of minimal charts and their applications VI: the graph $\Gamma_{m+1}$ in a chart $\Gamma$ of type $(m;2,3,2)$

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 $(m;2,3,2)$ if $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=2$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=3$, 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 if there is a minimal chart $\Gamma$ of type $(m;2,3,2)$, then each of $\Gamma_{m+1}$ and $\Gamma_{m+2}$ contains one of three kinds of graphs. In the next paper, we shall prove that there is no minimal chart of type $(m;2,3,2)$.