Planar graphs

Objective: This research was carried out in order to investigate the planar graphs and their generalizations to those on compact surfaces. Material and Methods: Planar graphs are graphs that can be drawn on a plane without any edge crossings. Planarity of graphs are characterized by Kuratowski's teorem. Nonplanar graphs can be drawn on surfaces of genus greater than with no edge crossings. Here the genus of the surfaces is determined by the Euler-Poincare formula. Results: In this thesis, mainly the graphs with the same vertex degree and the faces valency have been investigated. Such a graph is called -platonic. Conclusion: It is known that the number of -platonic graphs of genus and greater than is finite, and that of genus is infinite. In this thesis, it has been shown that three different types of -platonic graphs can be embedded on tori. Also, for every , five -platonic grahp family, which are contained by three different surfaces, have been introduced. Key Words: Genus of a Graph. Graph, Planar Graph, Platonic Graph.