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 $(m;2,3,2)$.

39 pages, 31 figures. arXiv admin note: text overlap with arXiv:1902.00007, arXiv:1603.04639, arXiv:1609.08257