Planar Graphs 1 / 28 Planar Graphs Definition A graph is planar if it can be drawn in the plane with no edges crossing. 2 / 28 Examples of Planar Graphs C4 and K4 are both planar. 3 / 28 Proof Idea Remove edges until no cycles exist, always maintaining connectivity. Each removal of an edge from a cycle reduces the number of edges by 1 and the number of regions by 1, and leaves the number of vertices unchanged. The resulting graph is a tree, which has one less edge than vertices, and which has one region. Euler's Formula Theorem For any connected plane graph with p vertices and q edges that divides the plane into r regions, we have p − q + r = 2: 4 / 28 Euler's Formula Theorem For any connected plane graph with p vertices and q edges that divides the plane into r regions, we have p − q + r = 2: Proof Idea Remove edges until no cycles exist, always maintaining connectivity. Each removal of an edge from a cycle reduces the number of edges by 1 and the number of regions by 1, and leaves the number of vertices unchanged. The resulting graph is a tree, which has one less edge than vertices, and which has one region. 5 / 28 Regional Degree Theorem Definition The degree of a region of a plane graph is the number of encounters with edges in a walk around the boundary of the region. Theorem For any connected planar graph with q edges that divides the plane into r regions, we have 2q = sum of degrees of the regions : 6 / 28 Example of a Non-Planar Graph Consider a connected graph with p = 6 vertices and q = 13 edges. If the graph were planar, then by Euler's formula it would have r = 9 regions. c b a f d e 7 / 28 K5 is not planar K5 has p = 5 vertices and q = 10 edges. If K5 were planar, it would have r = 7 regions. b a c e d 8 / 28 K3;3 is not planar K3;3 has p = 6 vertices and q = 9 edges. If K3;3 were planar, it would have r = 5 regions. a b c d e f 9 / 28 Petersen graph is not planar Petersen graph has p = 10 vertices and q = 15 edges. If Petersen graph were planar, it would have r = 7 regions. g h b c a f d e i j 10 / 28 What makes a graph non-planar? I Does having large cycles make a graph non-planar? I Does having many edges make a graph non-planar? I Does having a non-planar subgraph make a graph non-planar? 11 / 28 Planarity of Complete Graphs Theorem Kn is planar if and only if n ≤ 4. 12 / 28 Planarity of Complete Bipartite Graphs Theorem Km;n is planar if and only if min(m; n) ≤ 2. 13 / 28 Is this graph non-planar? c b a f d e 14 / 28 Subdivision of graphs Definition Inserting a new vertex into an existing edge of a graph is called subdividing the edge. One or more subdivisions of edges creates a subdivision of the original graph. 15 / 28 Kuratowski's Theorem Theorem A graph is planar if and only if it does not contain a subdivision of K5 or K3;3 as a subgraph. 16 / 28 Planarity Algorithm for Hamiltonian Graphs Let G be a Hamiltonian graph. 1. Draw the graph G with Hamilton cycle H on the outside, i.e. with H as the boundary of the infinite region. 2. List the edges of G not in H: e1;:::; er . 3. Form a new graph K in which the vertices are labelled e1;:::; er and where the vertices labelled ei ; ej are joined by an edge if and only if ei ; ej cross in the drawing of G, i.e. cannot both be drawn inside (or outside) H (such edges are said to be incompatible). 4. Then G is planar if and only if K is bipartite. 17 / 28 Example Verify that the given graph has a Hamilton cycle and find a plane diagram for the graph. b c a d e 18 / 28 Example Hamilton cycle drawn on Corresponding incompatible the outside. edge graph. c ab ad a d ce bc b e 19 / 28 Convex Regular Polyhedra A polyhedron is convex when the straight line segment joining any two of its vertices lies entirely within it. A convex polyhedron can be represented by a planar graph. A polyhedron is regular when there exist integers m ≥ 3, n ≥ 3 such that each vertex has m edges meeting at it, and each face has n edges on its boundary. 20 / 28 Tetrahedron Edges: 6; Vertices: 4; Regions: 4 21 / 28 Cube Edges: 12; Vertices: 8; Regions: 6 22 / 28 Octahedron Edges: 12; Vertices: 6; Regions: 8 23 / 28 Dodecahedron Edges: 30; Vertices: 20; Regions: 12 24 / 28 Icosahedron Edges: 30; Vertices: 12; Regions: 20 25 / 28 Classification of Convex Regular Polyhedra Theorem Suppose that a regular polyhedron has each vertex of degree m and each face of degree n. Then (m; n) is one of (3; 3), (3; 4), (4; 3), (3; 5), (5; 3). Further, there exist Platonic solids corresponding to each of these pairs. 26 / 28 Platonic Solids Corresponding to (m; n) m n q p r Name 3 3 6 4 4 tetrahedron 3 4 12 8 6 cube 4 3 12 6 8 octahedron 3 5 30 20 12 dodecahedron 5 3 30 12 20 icosahedron 27 / 28 Acknowledgements Statements of results follow the notation and wording of Anderson's First Course in Discrete Mathematics. 28 / 28.