Upper Bound On The Demigenus Over All Signatures Of K_4,n

Find the largest demigenus over all signatures on the complete bipartite graph nmK,is an open question raised by Achdeacon. In this paper, we first determine the switching equivalent classes and demigenus of orientation embedding over all signatures on4,4K and5,4K.Based on this,by the method of diamond product and combining induction, we determined the maximum or the upper bound of all signature on the complete bipartite graphnK,4.This paper is mainly divided into the following three parts.In chapter 1, this paper introduces the background of graph theory and signed graph, generally introducing the present situation and the main problems of it’s research.In chapter 2, it clarifies the related concepts and some important lemmas of signed graph.In chapter 3, it determined the demigenus over all signature on the complete bipartite graph 4,4K and 5,4K.In chapter 4, by the method of diamond product and combining induction, we proof that the largest demigenus over all signature onnK,4is no more than n.