Decomposition and 4-colouring of apair of dual trees of Heawood graph

This paper clarifies the basic concept concerning the 4-colouring of arbitrary plannar graph,and gives a definition of dual trees.A pair of dual trees in the dual graph and the Hamilton path on the dual graph are interdependent,and so a kind of 4-colouring of arbitrary plannar graph is given.Based on the above mentioned theory,this paper gets a pair of dual trees and its property.And then, the origin and basic characteristics of Heawood graph are describled,and two methods and steps of 4-coloring of Heawood graph are introduced.The 4-coloring of Heawood graph is implemented by dividing the dual graph into two regions.With the decomposition of the Heawood graph of Hamilton path of the dual graph,two pairs of dual trees are constructed.236 different 4-colouring schemes on 25 vertices of Heawood graph obtained by author's method are given,consequently the flaw in Kempe's 4-colouring conjecture proof is remedied and Heawood graph is 4-colourable.