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

Let Gamma be a chart, and we denote by Gamma(m) the union of all the edges of label m. A chart G is of type (3, 2, 2) if there exists a label m such that w(Gamma) = 7, w(Gamma(m) boolean AND Gamma(m+1)) = 3, w(Gamma(m+ 1) boolean AND Gamma(m+2)) = 2, and w(G Gamma(m+ 2) boolean AND 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).