| Since the introductory investigation of maximum genus by Nordhaus, Stewart, and White[1], the upper embeddability of graphs has been developing so far. Although Xuong[2],Liu[3] and Nebesky[4] have respectively provided different necessary and sufficient conditions on the upper embeddability of graphs,we know less about what classes of graphs are upper embeddable.If the upper bound on the maximum genus of a graph G can arrive at the best value [β(G)/2], then we call such a graph G is upper embeddable. In recent years, many graphs theory scholars research on the relations between parameters and the upper embeddability of graphs. They have attained many classes of upper embeddable graphs. In this paper we provide some new classes of upper embeddable graphs by combining with the max degree of vetex in cycle,the dominate vertices set of graphs, and the contidions of A-partition. The details are as follows.In chapter one and two, we introduce the background and required knowledge of the upper embeddability of graphs, and give primary conceptions and theorems .In chapter three, combining with the max degree of vetex in cycle,we attain two classes of upper embeddable graphs.In chapter four, combining with the dominate vertices set of graphs,we attain two classes of upper embeddable graphs.In chapter five, combining with A-partition , degree of vertex and edge-connectivity, we attain some new classes of upper embeddable graphs. Thus we generalize further the relative results, and characterize entirely such upper embeddable graphs.In the end ,we introduce the researching prospect of the upper embeddability of graphs and state author's next work. |
