K4

Definition

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.

5 Must Know Facts For Your Next Test

1. The graph k4 contains exactly 4 vertices and 6 edges, demonstrating its completeness.
2. k4 is not a planar graph because it cannot be drawn on a plane without edges crossing each other.
3. In terms of connectivity, k4 is maximally connected, meaning that removing any single vertex still leaves the graph connected.
4. k4 serves as a critical example when discussing Kuratowski's theorem, which identifies forbidden subgraphs for planar graphs.
5. In applications, k4 can represent fully connected networks with four nodes, useful in various fields including computer science and network theory.

Review Questions

• How does k4 exemplify the characteristics of complete graphs and what does this imply about its connectivity?
  • k4 exemplifies complete graphs by having every vertex connected to every other vertex, resulting in 6 edges for 4 vertices. This characteristic implies that k4 is maximally connected; even if one vertex is removed, the remaining three vertices still maintain connections among themselves. This high level of connectivity makes k4 a useful model in network theory for understanding fully connected networks.
• Discuss why k4 is not considered a planar graph and relate this to Euler's Formula.
  • k4 is not considered a planar graph because it cannot be drawn on a flat surface without edges crossing. According to Euler's Formula, for a planar graph V - E + F = 2 must hold true. In the case of k4 with V = 4 and E = 6, if we assume it could be planar, we would require F to equal 4, which leads to inconsistencies with the formula. Hence, k4 serves as an important example of a non-planar structure.
• Evaluate the significance of k4 within Kuratowski's theorem and its implications for identifying non-planarity in graphs.
  • Kuratowski's theorem states that a finite graph is non-planar if and only if it contains a subgraph that is a subdivision of either k5 or k3,3. Since k4 is not itself non-planar but indicates properties related to subgraphs, recognizing its structure helps identify potential non-planarity in larger graphs. Understanding k4’s role within this framework aids in the analysis of more complex graphs that could contain configurations leading to non-planarity.

Related terms

Complete Graph: A complete graph is a type of graph where every pair of vertices is connected by a unique edge, denoted as Kn, where n is the number of vertices.

Planar Graph: A planar graph is a graph that can be drawn on a plane without any edges crossing, which plays a crucial role in understanding k4 and its implications in Euler's Formula.

Euler's Formula: Euler's Formula relates the number of vertices (V), edges (E), and faces (F) in a connected planar graph through the equation V - E + F = 2.