Research On The Anti-magic Conjecture Of Graphs

Graph theory is an important part of discrete mathematics,It is an important branch of modern applied mathematics,Graph theory is widely used in many fields,such as physics,chemistry,operational research,computer science,information theory,cybernetics,network communication,social science and military,national defense,economic management,industrial and agricultural production.In this paper,we mainly researched the two antimagic labeling problems which are the graph of direct products and the join of graphs.The antimagic labeling problems derived from the conjectures proposed by Hartsfield and Ringel.The graph labeling of G=(V,E),is a bijective mapping:?:E?{1,2,3,…,|E|}.For each vertex u of G,the vertex-sum at u is defined as f?(u)=?e?E(u)?(e),where E(u)is the neighbourhood of u and it is the set of edges incident to u.Hartsfield and Ringel had conjectured that every connected graph other than K2 is antimagic.And the two scholars had turned out that Pn,Sn,Cn,Km,Wm,m?3 are antimagic.In the progress of study antimagic labeling problems,the arrangement of numbers has been widely used to label the circle in this paper.In the meantime,we adopted the ways of different combinations of numbers and generalized magic matrix to solve the conflicts which are the two different vertices have the same vertex-sum.And finally,we gained that the direct product of path and circle,direct product of star and path,direct product of wheel and path,the join of circle and many disjoint circles graph are antimagic.The results which obtained in this paper are compensations in the field of antimagic labeling of graphs,in the meantime it comprehensive,improved and extended the latest researches which gained by many scholars.The paper has four parts:the first one is introduction.In this section,we introduced the development history of graphs and some basic conceptions.The second part,we studied the antimagicness of the direct product which are path and circle,star and path,wheel and path.Firstly,we show the two methods which gained in the process of labeling,after that we raised the two ways of modification to solve the conflicts which belong to vertex-sum,Respectively to prove direct product of path and circle,direct product of star and path,direct product of wheel and path are antimagic.The third part,we study the join of circle and many disjoint circles graph is antimagic.First of all,we introduced the conception of the join graph and the labeling techniques.The next is to show generalized magic matrix.Finally,prove that he join of circle and many disjoint circles graph is antimagic.In the last part is summary and expectation.In this part,w e obtained the conclusion that the direct product of path and circle,direct product of star and path,direct product of wheel and path,the join of circle and many disjoint circles graph are antimagic.And looking forward to the work which we will do in the future.