On The Adjacent Strong Edge Coloring And 2-strong Edge Coloring Of Some Graphs

The coloring problems is one of popular problems in graph theory because of it's profound significance in both theory and practice.Many issues in discrete systems can be translated into the problems of graph coloring.For example,the edge maximum of n-order graph which don't contain a certain graph G as a subgraph depends on the chromatic number of the graph.Therefore T R.Jensen and B.Toft asserted:the graph coloring theory in discrete mathematics at the center position.In real life,many areas will be dealt with the object of a certain set of rules according to certain classification of the problem.For example,schedule,scheduling,time-table problem,the problem of storing,circuit arrangement,task allocation,and so on.These problems are closely related to coloring theory.The so-called graph coloring is refers speaking of the graph in the vertex,edge(to the plane graph also has face) and so on the element carries on the classification according to the certain rule.Object dissimilarity or rule dissimilarity,then has all kinds of colotings.With the development of coloring theory,there are many new coloring, such as total coloring,list coloring,strong edge coloring,strong coloring,choosable coloring,r-strong edge color and so on hundred and thousand of colouring ways.These has bscome new hot spots in the graph colouring.As to the unresolved questions in the past,we can put them into a new coloring problem such that it is easy to understand and convenient to study.In this paper,we discuss the adjacent strong edge coloring and 2-strong edge coloring of some graphs.First,with the research of 2-strong edge coloring and 3-strong edge coloring of tree from Akbari and adjacent strong edge coloring of tree from Zhang Zhongfu,a sufficient condition is proposed that 2-strong edge coloring number equals to the maximum degree plus one.Then according to the struct of cycle,complete graph, complete bipartite graph,wheel,nacklace,P_n~2,P_n~3,P_n~4,P_n~5,monocycle graph,some join-graphs,square mesh,hexagonal mesh and the relationship between the vertexes with the maximum degree,the 2-strong edge coloring of these graphs is studied to obtain the relationship of 2-strong edge coloring number and the maximum degree. Besides,the coloring method on the 2-strong edge coloring of these graphs are given.