On The Maximum Genus Of Graphs

The concept of the embeddablility of graph comes from the planarity.In the beginning of 30's,a Polish mathematician K.Kuatowski and mathematician of the United States H.Whitney,S.Maclane did the consummate research in the embed-dablility of graph.They found the theory by themself.In the 50's,Wenjun WU the mathematician of china,proved a rule about the planarity of graph based on the algebra topology.And some scholars later ameliorated this result and brought for-ward the better arithmetic.Yanpei[1]Liu made complicacy of arithmetic on the top.It made great contribution in planarity and embeddability of graph.Through research of the problem,in the 70's,Nordhuas E,Stewart B,White A[2]brought for-ward the problem of graph embedding on the surface.And the maximum genus and up-embeddability of a graph are the important parts about the graph embed-ding on the surface.The genus is a topological invariant.This text sum up and research maximum genus of the graph which has some properties or any one character.The all text is partitioned to three portions.In chapter 1,first there are some essential knowledge about graph including the theory of embeddability of graph.From the graph embedding on plane to sur-face,the text mostly depicts the theory of maximum genus.It includes two essen-tial theory,one is Xuong's theory[4],the other is Nebesk(?)'s theory[4].They both use the parameter Betti deficiency to confirm the computing method of maximum genus.Subsequently according to the Nebesk(?)'s theory[5],Yuanqiu Huang[6]pro-vides a connection between Betti deficiency and structural character of graph.It inaugurates a new way of research of maximum genus from the structural character of graph.In chapter 2,some results that have be known about maximum genus and parameters of graph could be recapitulated.These parameters include degree of vertex,connectivity of graph,diameter,cut-vertices,independence number,degree of embeddable region,color number,regular,2-factor and so on.About diameter Yuan-qiu Huang and Yanpei Liu[10]prove the lower bound of maximum genus of graph G with diameter four not containing complete subgraph K₃.But this result is not sharp.In the second section,the text improve the result.I have published a article[29].In the article,it shows that the result is sharp.Last,for the discretional finite undirected acyclic graph,a method of constructing up-embeddable graph is provided.In chapter 3,it is tag.There are some my opinions about the maximum genus.I hope that it could provide a little help for the work of research in the future.