Some Problems In Topological Graph Theory

At present, topological graph theory is a very active component of graph theory in the world, and investigating the topological parameter of graph(the genus of graph) is a very important subject. It is a characteristic parameter that determines whether a graph has a 2-cell embedding in a certain orientable surface. It is known from the interpolation theorem of the genus of graph, we only consider the minimum and the maximum genus of graph. In this paper, we mainly investigate the two parameters and obtains follow results:(1) Let G be a graph, and letω(G) be the edge covering number of G, and g(G) be the girth of G. Combining with the conditions of the edge covering number and girth, we obtain a upper bound of Betti deficiencyξ(G), and we give a lower bound on the maximum genus.(2) Combining with the condition of 4-quadrangle 2-factor, we give classes of upper embeddable graphs. Based on the known results, we characterize entirely the upper embeddability of such classes of graphs containing a 4-quadrangle 2-factor.(3) By investigating the relationship between the upper embeddability of graphs and the diameter of G, we obtain some new upper embeddable graphs, and then we can complement some recent results of the upper embeddable graphs about the diameter of G.(4)With the help of the theory of embeddings of a voltage graph and its cover graph, we obtain the genus of a class of tensor product graph. Meanwhile, we further generalize some results in recent papers.