K3,3

Definition

The term k3,3 refers to a specific complete bipartite graph that has two sets of vertices, each containing three vertices, and is denoted as K_{3,3}. In this graph, every vertex from one set is connected to every vertex in the other set, making it an important example in the study of graph theory and planar graphs. This graph is significant for demonstrating concepts related to non-planarity and graph coloring, showcasing how certain configurations cannot be drawn on a plane without edges crossing.

5 Must Know Facts For Your Next Test

1. The k3,3 graph is non-planar, which means it cannot be drawn on a plane without some edges crossing.
2. It serves as a fundamental example in Kuratowski's theorem, which states that a graph is non-planar if and only if it contains a subgraph that is a subdivision of either k3,3 or K5.
3. In terms of edges and vertices, k3,3 has a total of six edges connecting the vertices from one set to the other.
4. The minimum number of colors needed to color k3,3 is two, illustrating its bipartite nature.
5. k3,3 has applications in various fields including network theory and computer science, particularly in problems involving matching and scheduling.

Review Questions

• How does the structure of k3,3 illustrate the concept of bipartite graphs?
  • The structure of k3,3 demonstrates the concept of bipartite graphs because it consists of two distinct sets of vertices, with three vertices in each set. In this configuration, each vertex in one set connects to every vertex in the other set but has no internal connections within its own set. This property aligns perfectly with the definition of bipartite graphs and helps in understanding how different types of graphs can be structured.
• Discuss the significance of k3,3 in relation to Kuratowski's theorem and its implications for planar graphs.
  • Kuratowski's theorem highlights that k3,3 is a critical example of non-planarity. According to this theorem, a graph is non-planar if it contains k3,3 or K5 as a subgraph. This establishes k3,3 as a fundamental case study in understanding planar graphs. Recognizing that k3,3 cannot be embedded in a plane without edge crossings helps mathematicians identify non-planar structures and their properties.
• Evaluate how the characteristics of k3,3 can be utilized in real-world applications such as network design or scheduling problems.
  • The characteristics of k3,3 are highly relevant in real-world applications like network design and scheduling problems due to its bipartite nature. In networking scenarios, it can represent relationships between two distinct groups where each member from one group needs to connect with all members from another group. This allows for optimal resource allocation or task assignment. The principles derived from studying k3,3 also assist in formulating algorithms for matching problems where efficient connections between two sets are required.

Related terms

Bipartite Graph: A type of graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.

Planar Graph: A graph that can be drawn on a plane without any edges crossing each other.

Graph Coloring:

An assignment of labels or colors to the vertices of a graph such that no two adjacent vertices share the same color.