Petersen Graph

Definition

The Petersen graph is a well-known, highly symmetrical graph that has 10 vertices and 15 edges. It is often used in combinatorics due to its interesting properties, such as being 3-regular and non-bipartite. The Petersen graph serves as an important example for various graph theoretical concepts, illustrating the characteristics of special types of graphs, including their connectivity and chromatic number.

5 Must Know Facts For Your Next Test

1. The Petersen graph has a chromatic number of 3, meaning it requires three colors to color its vertices without adjacent vertices sharing the same color.
2. It is not a bipartite graph, as it contains odd-length cycles, which disqualifies it from being divided into two independent sets.
3. The Petersen graph is symmetric and can be represented in multiple ways while maintaining its structural properties.
4. Every vertex in the Petersen graph connects to exactly three other vertices, making it a classic example of a regular graph.
5. The Petersen graph is also known for its role in demonstrating various properties of other graphs, such as in discussions about graph embeddings and embeddings in surfaces.

Review Questions

• How does the structure of the Petersen graph illustrate the concept of a 3-regular graph?
  • The Petersen graph is a perfect example of a 3-regular graph because each of its 10 vertices is connected to exactly three other vertices. This uniform degree distribution shows the regularity across the graph, which helps in understanding how regular graphs function within the larger context of graph theory. The equal connectivity among vertices facilitates various analyses related to symmetry and vertex relationships.
• What makes the Petersen graph non-bipartite, and how does this characteristic affect its chromatic number?
  • The Petersen graph is non-bipartite because it contains odd-length cycles, specifically a cycle of length 5. This characteristic directly impacts its chromatic number, which is 3; meaning that at least three colors are required to color its vertices such that no two adjacent vertices share the same color. In bipartite graphs, two colors are sufficient due to their lack of odd-length cycles.
• Evaluate the significance of the Petersen graph in combinatorics and how it can be used to illustrate broader concepts within the field.
  • The Petersen graph holds significant value in combinatorics as it serves as a counterexample for many conjectures related to regular graphs and colorings. It demonstrates properties like non-bipartiteness and symmetry that can be pivotal in understanding complex interactions between vertices in various types of graphs. Additionally, its unique characteristics make it useful for illustrating broader concepts such as graph embeddings, connectivity, and even applications in network design and error-correcting codes.

Related terms

3-Regular Graph: A graph where every vertex has exactly three edges connected to it.

Graph Isomorphism:

A bijective mapping between the vertex sets of two graphs that preserves the edge connections.

Chromatic Number: The smallest number of colors needed to color a graph's vertices so that no two adjacent vertices share the same color.