7.1 Planar Graphs and Euler's Formula
3 min read•august 12, 2024
Planar graphs are the rock stars of graph theory. They can be drawn on a flat surface without any crossing. This concept is crucial for understanding how networks and connections work in the real world.
is the secret sauce that ties it all together. It shows a simple but powerful relationship between the number of , edges, and faces in a . This formula unlocks insights into graph structure and properties.
Planar Graphs and Embeddings
Understanding Planar Graphs and Their Components
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- Planar graph represents a graph drawn on a plane without edge crossings
- refers to the specific arrangement of a planar graph on a two-dimensional surface
- Vertices function as points or nodes in the graph, representing entities or intersections
- Edges connect vertices, symbolizing relationships or connections between nodes
- encompass areas bounded by edges, including the infinite outer region
Visualizing Planar Graphs
- planar graphs involves strategic placement of vertices and edges
- Multiple valid embeddings can exist for a single planar graph
- (complete graph with 4 vertices) serves as a simple planar graph (can be drawn without crossings)
- (complete graph with 5 vertices) cannot be drawn as a planar graph (always requires edge crossings)
- Real-world applications include circuit board design and transportation networks
Faces and Euler's Formula
Understanding Faces in Planar Graphs
- represents a region bounded by edges in a planar graph embedding
- Faces include both finite interior regions and the infinite exterior region
- Number of faces varies depending on the specific embedding of the graph
- consists of three edges forming a closed loop (common in many planar graphs)
- refers to the number of edges surrounding it
Exploring Euler's Formula
- Euler's formula establishes a fundamental relationship between vertices, edges, and faces
- Formula states: , where V = number of vertices, E = number of edges, F = number of faces
- Applies to all , including trees and cycles
- Provides a method to calculate one component given the other two (V, E, or F)
- Generalizes to other surfaces (torus, sphere) with modified constants
Applications of Euler's Formula
- Proves the existence of only five (regular polyhedra)
- Helps determine the maximum number of edges in a planar graph with n vertices
- Useful in analyzing map colorings and solving related graph theory problems
- Applies to the study of and molecular geometry
- Facilitates the analysis of polyhedra and their properties in geometry
Advanced Topics
Exploring Dual Graphs
- transforms faces of the original graph into vertices
- Edges in the dual graph connect adjacent faces from the original graph
- Preserves planarity, meaning the dual of a planar graph remains planar
- have isomorphic dual graphs (platonic solids exhibit this property)
- Applications include optimizing network flows and solving maze problems
Understanding Kuratowski's Theorem
- provides a characterization of planar graphs
- States that a graph is planar if and only if it contains no subdivision of K5 or
- K5 represents the complete graph on 5 vertices
- K3,3 denotes the complete bipartite graph with two sets of 3 vertices each
- Theorem offers a systematic method for determining graph planarity
- Subdivisions involve adding vertices along edges without changing graph structure
- Plays a crucial role in graph minor theory and topological graph theory
Key Terms to Review (21)
Chemical Structures: Chemical structures refer to the arrangement of atoms within a molecule, including the bonds that hold them together and their spatial orientation. This concept is crucial in understanding the properties and behaviors of different substances, as the structure of a compound directly influences its reactivity, stability, and interactions with other compounds. In the study of planar graphs and Euler's formula, chemical structures can be represented as graphs where atoms are vertices and bonds are edges, illustrating relationships and properties in a visually meaningful way.
Connected Planar Graphs: Connected planar graphs are a type of graph that can be drawn on a plane without any edges crossing and where there is a path between any two vertices. This property of being connected ensures that every vertex is reachable from any other vertex, while planarity emphasizes the graph's ability to maintain its structure without overlapping lines. Together, these characteristics are fundamental when exploring graph theory, especially in relation to Euler's formula, which relates the number of vertices, edges, and faces in a planar graph.
Degree of a face: The degree of a face in a planar graph refers to the number of edges that are incident to that face. This measure helps in understanding the relationships between the faces, vertices, and edges in the graph. By analyzing the degree of faces, one can draw conclusions about the overall structure of the graph, including aspects such as connectivity and planarity, which are central to various geometric theories.
Drawing: In the context of planar graphs, a drawing refers to a visual representation of a graph in a plane where the vertices are represented as points and the edges as curves connecting these points. A good drawing does not allow edges to cross each other, which is crucial for understanding the properties of planar graphs. The ability to create such drawings is essential for applying Euler's formula and examining the relationship between vertices, edges, and faces in planar graphs.
Dual graph: A dual graph is a graph that represents the relationships between the faces of another graph, where each vertex of the dual graph corresponds to a face of the original graph, and each edge represents the adjacency between two faces. This concept is crucial for understanding properties of planar graphs, as well as the relationship between Voronoi diagrams and Delaunay triangulations, where each structure can be seen as a dual of the other.
Edges: Edges are the line segments that connect vertices in a graph, forming the structure of the graph itself. They play a crucial role in defining relationships between vertices, determining connectivity, and influencing properties such as path lengths and cycles. In the context of geometric structures, edges can also represent the boundaries of shapes, like the sides of polygons or polyhedra.
Embedding: Embedding refers to the way a graph is drawn on a surface without any edges crossing each other, preserving the graph's structure. This concept is critical when analyzing planar graphs, as it not only defines how these graphs can be visually represented but also helps in understanding their properties and relationships with surfaces. An embedding of a graph shows how vertices and edges can be arranged in a two-dimensional space while ensuring that the connections remain intact and do not overlap.
Euler's Formula: Euler's Formula is a fundamental equation in geometry that relates the number of vertices (V), edges (E), and faces (F) of a convex polyhedron through the equation $$V - E + F = 2$$. This relationship highlights the intrinsic link between these geometric components and serves as a cornerstone in various branches of mathematics, particularly in discrete geometry and topology.
Face: In geometry, a face refers to any of the flat surfaces that make up a polytope. Faces can be two-dimensional shapes that form the boundaries of three-dimensional objects and play a critical role in understanding the structure and properties of polytopes. They can vary in shape and size, and the study of faces includes investigating how they relate to one another and contribute to the overall geometric characteristics.
K3,3: k3,3 is a specific type of bipartite graph consisting of two sets of vertices, each containing three vertices, with edges connecting every vertex in one set to every vertex in the other set. This graph is significant in the study of planar graphs because it serves as a classic example of a non-planar graph, meaning it cannot be drawn on a plane without edges crossing. Understanding k3,3 helps to illustrate key concepts related to Euler's formula and the properties of planar graphs.
K4: k4 is a complete graph on four vertices, meaning that every pair of distinct vertices is connected by a unique edge. This graph serves as an important example in the study of planar graphs, as it is one of the simplest structures to demonstrate properties like connectivity and can be used to understand more complex planar graph behaviors.
K5: k5 is a complete graph consisting of five vertices, where every pair of distinct vertices is connected by a unique edge. This graph serves as an important example in the study of planar graphs, as it is one of the first known graphs that is non-planar, meaning it cannot be drawn on a plane without edges crossing. Understanding k5 helps to illustrate concepts related to graph theory, particularly in connection with Euler's formula and the criteria for planar graphs.
Kuratowski's Theorem: Kuratowski's Theorem states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph $K_5$ (five vertices, all connected) or the complete bipartite graph $K_{3,3}$ (two sets of three vertices, each connected to every vertex in the other set). This theorem serves as a foundational principle in understanding planar graphs, connecting geometric properties with topological structures.
Map coloring: Map coloring is the process of assigning colors to regions of a map so that no two adjacent regions share the same color. This concept is essential in various fields, including graph theory and combinatorics, as it provides insights into problems related to resource allocation and scheduling. Map coloring often uses planar graphs, where each region can be represented as a vertex and edges connect vertices that are adjacent on the map.
Planar Graph: A planar graph is a graph that can be drawn on a plane without any edges crossing each other. This concept is crucial as it connects various mathematical principles, like Euler's formula, which relates the number of vertices, edges, and faces in a connected planar graph. Understanding planar graphs also involves techniques for testing their planarity and finding embeddings, as well as applications in polygon triangulation for efficient computational geometry.
Platonic Solids: Platonic solids are convex polyhedra with identical faces composed of congruent convex regular polygons. There are exactly five such solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. These shapes are fundamental in geometry, as they represent the only regular polyhedra that can exist in three-dimensional space, and they have significant applications in various fields like crystallography and architecture.
Regions: In discrete geometry, a region is defined as a connected subset of a space that is bounded by certain geometric properties. Regions can be thought of as the areas enclosed by curves or surfaces, and they play a crucial role in understanding the structure and relationships within geometric figures, especially in the context of planar graphs and Euler's formula.
Self-dual graphs: Self-dual graphs are graphs that are isomorphic to their dual graphs. In simpler terms, if you take a planar graph and create a new graph by swapping faces and vertices, a self-dual graph will look the same as the original. This property connects them to important concepts like planarity, Euler's formula, and the relationships between vertices, edges, and faces in a graph.
Triangular face: A triangular face is a flat surface of a three-dimensional geometric shape that has three edges and three vertices, forming a triangle. Triangular faces are fundamental components of polyhedra, particularly in structures such as tetrahedra and triangular prisms, and they play a crucial role in determining the properties and characteristics of these shapes.
V - e + f = 2: The equation v - e + f = 2, known as Euler's formula, is a fundamental relationship in graph theory that applies to planar graphs. In this equation, 'v' represents the number of vertices, 'e' signifies the number of edges, and 'f' denotes the number of faces in a connected planar graph. This relationship reveals essential properties of planar graphs and helps to establish a deeper understanding of their structure.
Vertices: Vertices are the distinct points in a geometric figure where two or more edges meet, often serving as the corners or intersections of shapes. In various contexts, such as graph theory and polytope geometry, vertices play a crucial role in defining the structure and properties of these figures, impacting their connectivity and dimensional characteristics.