Embedding Bipartite and Complete Graphs on Orientable and Non-Orientable Surfaces

Abstract

In this paper we will observe the progress having been made in the embedding of regular euler impure graph embeddings on orientable and non-orientable surfaces including the torus T2, real projective plane (RP2), and klein bottle where regular graphs are a graph with a set of nodes where every node can connect to every other node denoted by Km. We will also discuss a partition to be made within an infinite family of euler impure graphs on orientable surfaces as given by Davies and Pfender. We will do so by considering natural solutions to the g/2 floor stipulation. After which, we will discuss constructions of bipartite euler impure graph embeddings on orientable and non-orientable surfaces and their properties where bipartite graphs are graphs with 2 sets of nodes that can connect to every node in the other set denoted by K(m,n). We will also discuss arbitrary node embeddings on the 4-faces of these bipartite euler-impure graph constructions. This construction is heavily inspired by the methodology toward the regular case in the paper,”Edge-maximal graphs on orientable and some non-orientable surfaces,” written by Davies and Pfender.