Characterization Of Some Properties Of Ribbon Graphs And Their Partial Duals

A ribbon graph is a surface with boundary and a cellularly embedded graph can be realized as a ribbon graph.The concept of partial dual generalizes the fundamental concept of the geometric dual of a cellularly embedded graph.It was introduced to unify various versions of Thistlethwaite theorems in knot theory that relate the Jones polynomial of knots with a Tutte-like polynomial of graphs.Partial duality is far-reaching extensions of geometric duality and has found a number of significant applications in graph theory,topology,and physics.In this dissertation,we focus on Eulerian and even-face characterizations of rib-bon graphs and their partial duals.The dissertation consists of five chapters.In Chapter 1,we summarize the background of the field,and state the main results of the present thesis,and give the outline of this dissertation.In order to discuss questions conveniently we also give some basic knowledge in this chapter.Chapter 2 consists of the following:definition of ribbon graphs and its equiv-alence of cellularly embedded graphs;advantages of ribbon graphs over cellularly embedded graphs;definitions of some new concepts such as line-segments,vertex-line-segments and edge-line-segments;importance of partial duals;definition of arrow marked ribbon graphs and pictorial illustration of taking a partial dual.It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite.In 2013 Huggett and Moffatt extended this result to partial duals of plane graphs and then characterized all bipartite partial duals of a plane graph using all-crossing directions of its medial graph.They left the characterization of all Eulerian partial duals of a plane graph as an open problem.In Chapter 3 we solve this problem by considering half-edge orientations of medial graphs and allowing some inconsistent edges.In Chapter 4 we further study this problem and generalize the above well-known theorem to embedded graphs and partial duals of cellularly embedded graphs,and characterize all Eulerian and all even-face(i.e.a cellularly embedded graph with no odd degree faces)partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.One of the deepest results in graph theory,the Robertson-Seymour Theorem(graph minor theory),is that graphs are well-quasi-ordered under the graph minor re-lation.This theorem may be reformulated as stating that every minor-closed family of graphs is characterized by a finite set of excluded minors.Although we know minor-closed families can be characterized by excluded minors,very few explicit characteri-zations are known.Perhaps the best-known is Wagner's Theorem which characterizes planar graphs as those with no K5 or K3,3 minors.In 2016 Chudnovsky et al.mod-ified the classical minor operation to introduce a notion of bipartite minors,which is a closed operation of bipartite graphs,and proved a bipartite analog of Wagner's the-orem:a bipartite graph is planar if and only if it does not contain K3,3 as a bipartite minor.In Chapter 5,motivated by the above paper,we introduce a special kind of mi-nor operation called an Eulerian-minor such that it keeps Eulerian characteristics of Eulerian graphs and characterize Eulerian graphs and planar Eulerian graphs in terms of excluded Eulerian-minors.We also consider two special kind of minor operations of ribbon graphs such that they keep Eulerian or even-face characteristics of ribbon graphs,and then characterize Eulerian,even-face ribbon graphs and plane Eulerian,plane bipartite ribbon graphs using excluded such minors.