5-regular simple planar graphs and D-operations

Our goal is to prove a generating theorem for the class E5 of all 5-regular simple planar graphs. This is a progress report. First, we will see the general information from Euler’s formula and the Discharge Method. Second, the basic graph operation D-operation will be introduced. Third, there are two cases to be discussed separately. A graph either contains an edge that is a part of three distinct triangles or does not contain such an edge. Fourth, list irreducible subgraphs under D-operation for each case. Then, we investigate irreducible subgraphs under “multiple D-operations” for each case. The minimal graph in E5 is unique and is isomorphic to the graph of Icosahedron I by Euler’s formula. Let W be a graph obtained from the wheel W5 by removing an edge from the rim. We can prove that each graph in E5 contains W as a subgraph by the Discharge Method. Now, we will define a graph operation called D-operation, which is an H-type operation in Toida [1]. Let G be a graph in E5 in this article. A D-operation will be applied to adjacent vertices {x, y} or to an edge xy in G. Applying a D-operation to adjacent vertices {x, y} is to delete {x, y} from G and then reconnect neighbors of {x, y} so that a resulting graph is 5-regular. Note that applying D-operation to {x, y} is not unique. If you can obtain a resulting graph in E5 again, then we say that G is reducible by {x, y}, or the edge {x, y} is reducible. If any adjacent vertices in G is not reducible, then we say that G is irreducible (under D-operation). If G contains an edge xy that is a part of three distinct triangles, then we call xy a 3∆ edge and we say that G is in the 3∆ case. Otherwise, we call G non-3∆ case or 2∆ case since G contains W .