Edge Colouring Of The Planar Graph With Some Special Conditions

All of the graphs discussed in this paper are simple, undirected and finite. For one graph G=G(V(G),E(G)), V(G) and E(G) separately mean vertex set and edge set. For the vertex v∈V(G), d(v) is used to show the degree;△(G) andδ(G) each mean the maximum and minimum degree of the vertex in G, simply marked as△andδ. In graph theory symbols, the letters V,E,v andεare generally used to take the place of V(G), E(G), v(G) andε(G).The coloration of graphs is one of the main problems to be studied for graph theory. The graphs coloring can generally be divided into edge coloring, vertex coloring, total coloring and other specific colorings. This paper mainly discusses the edge coloring of planar graph and proves two main conclusions.A k-edge proper coloration of Graph G is a distribution of " k " kinds of colors on E(G) in such a way that the two adjacent edges are colored different colors. If there is a k-edge proper coloration of Graph G, then G is k-edge colorable. The edge chromatic number of graph G is the minimum value that makes G the k-edge colorable, marked as x (G). The following is the famous theorem for edge coloring.Vizing's theorem(1964) supposes that G is not a empty simple graph with no loop, then there is A(G)≤x'(G)≤A(G)+1.For the graph G with multiple edges, supposingμ(G) represents the max multiple number of each edge, then as a matter of fact, Vizing's theorem proves a more general conclusion:A(G)< x'(G)≤A(G)+μ(G).Vizing's theorem puts forward a classification problem:if G meets the requirement x'(G)=A(G), then G is called the graph of class one; if G satisfies x'(G)= A(G)+1, then it is called the graph of class two. It is very difficult to confirm whether a graph belongs to which class, and at present only a few graphs are clearly classified. The graph of class one includes path,tree and bipartite graph, even-order complete graph as well as wheels. Odd cycle and odd-order complete graphs are class two. Generally speaking, there still are not sufficient and necessary conditions to distinguish which class a graph belongs to. It is known that there are planar graphs of class two with the maximum degree 2,3,4,5. Meantime Vizing(1965) has proved that there is no graphs of class two with maximum degree above 8. Sanders and Zhao has proved that planar graphs with maximum degree 7 is the class one graphs with edge coloring. Now it is unknown to all of us whether there are the planar graphs of class two with the maximum degree 6.In this paper, the edge-coloring problem of planar graphs is mainly studied, and short cycle means these cycles that length is equal or lesser than 5. By redistributing the value of vertex and face and applying Euler's formula, the paper proves the following theorem:if G is a planar graph with△(G)=6, and there is no adjacent short cycles, then x'(G)=6; if G is a planar graph with A(G)=5, and there is no intersecting 3-cycle and short cycle, then x'(G)=5, that is to say, G is the graph of class one.