Complete Graphs, Bipartite Graphs, and Euler's Theorem in Discrete Mathematics

Explore complete graphs, bipartite graphs, and Euler's theorem in graph theory. This guide defines these graph types, discusses their properties, and provides examples illustrating these key concepts in discrete mathematics.

Complete Graphs, Bipartite Graphs, and Euler's Theorem

Complete Graphs

A complete graph is a graph where every vertex (point) is connected to every other vertex by an edge (line). A complete graph with n vertices is denoted as K_n.

Regular Graphs

A regular graph (or k-regular graph) is a graph where every vertex has the same degree (number of edges connected to it). A complete graph K_n is a regular graph of degree n-1.

Examples of Regular Graphs

(Illustrative figures would be included here showing examples of 2-regular and 3-regular graphs.)

Bipartite Graphs

A bipartite graph is a graph whose vertices can be divided into two disjoint sets (let's call them V₁ and V₂) such that every edge connects a vertex in V₁ to a vertex in V₂. There are no edges within V₁ or within V₂. A complete bipartite graph K_m,n has m vertices in V₁ and n vertices in V₂, with every vertex in V₁ connected to every vertex in V₂.

Examples of Bipartite Graphs

(Illustrative figures would be included here showing examples of K_2,4 and K_3,4.)

Complete Bipartite Graphs

In a complete bipartite graph, every vertex in one set is connected to every vertex in the other set. The number of edges is m * n.

Examples of Complete Bipartite Graphs

(Illustrative figures would be included here showing examples of K_3,4 and K_1,5.)

Euler Paths and Circuits

• Euler Path: A path that uses every edge of the graph exactly once.
• Euler Circuit (or Euler Cycle): An Euler path that starts and ends at the same vertex.
• Euler Graph: A graph that contains an Euler circuit.

Example of an Euler Graph and Circuit

(An illustrative figure of a graph would be included here, along with the Euler circuit highlighted.)

Euler's Theorem

Statement

For any connected planar graph G = (V, E, R), where V is the number of vertices, E is the number of edges, and R is the number of regions (faces), the following equation holds: V + R - E = 2

Proof (by induction)

The proof uses mathematical induction on the number of edges. The base case (one edge) and the inductive step (adding one edge) are described in the original text. (The figures illustrating the base case and inductive steps would be included here.)

Conclusion

Understanding complete graphs, bipartite graphs, and Euler's theorem is fundamental in graph theory. These concepts are used in many areas of computer science and mathematics to model and analyze networks and relationships.