enumeration of the number of spanning trees in some ... - Toubkal

30 CHAPTER 1. DEFINITIONS AND PROPERTIES1.1.2 Planar GraphsDefinition 1.1.29 A graph is planar if it can be drawn in the plane such that no edgesare crossing each other (its edges do not cross). This particular drawing of the graphis called a plane graph. For example K 4 is planar graph but K 3,3 is not planar graph.Although the complete graph with four vertices K 4 is usually pictured with crossing edgesas in Figure. 1.15(a), it can also be drawn with noncrossing edges as in Figure. 1.15(b);hence K 4 is planar.Figure 1.15: The graph K 4 drawn as a plane graph without edge crossing.Note that if G is disconnected, then G is planar if and only if each component is planar,hence we may assume well that G is connected throughout this thesis.Proposition 1.1.30 If H ⊆ G and H is not planar then neither is G. In particular K m,nis not planar if m, n ≥ 3.Definition 1.1.31 A planar graph partitions the plane into subsets called regions (faces).For example the plane graph of K 4 has four regions, one of which is exterior to the graph.1.2 MapsThe aim of this section is to provide a short and accessible presentation of planar maps.For a more detailed introduction, see the introductory chapter in [69] and the thesis of ElMarraki [43].1.2.1 Preliminaries and notations on mapsWe begin with some vocabulary on maps. A map is a proper embedding of a connectedgraph into the two-dimensional sphere, considered as continuous deformations. A map isrooted if one of its edges is distinguished as the root-edge and attributed an orientation.Unless otherwise specified, all maps under consideration in this thesis are rooted. Theface at the right of the root-edge is called the root-face and the other faces are said tobe internal. Similarly, the vertices incident to the root-face are said to be external andthe others are said to be internal. Graphically, the root-face is usually represented as theinfinite face when the map is projected on the plane.