Theorem on Regular Graph

1In this paper we rnean the gafigure consisting of simply connected polygons on a surface of genus zero1) Aside is one of the continuous lines common to distinct polygons and a vertex is a point common to three or more distinct polygons In 6ase a graph satisfies the following two conditions such a graph is called 7g17 (1) all vertices are triple (6belonging to three and only three polygons) and () every polygon has at least five sides Throughout this article we shall use the term graph for a regular graph Concerning the propery of a graph P Wernicke 2)has shown thaevery L graph must contain either (a) two adjacent pentagons or () apentagon adjacent to a hexagon 1Aft6rward this resUlt has been improved by P Franklin 3)thatis he has proved that every graph must contain either () apentagon adjacent to two other pentagons ) apentagon adjacent to a petagon and a hexagon or (c) apentagon adjacent to two hexagons The purpose of thispaper is to improve further their re3ults 0 Q In this paragraph we shall give a simple preliminary Denote byo1 2and the numbers of vertices sides polygons andsided polygons in a graph respectively By apPlying Euler-Poincar6 s formula 4)to a surface of genus zero we obtain the following relation (2D o212 From the hypothesis of regularity of such a graph we have (22) 3-2