K5

Definition

The term k5 refers to a complete graph on five vertices, denoted as K5, where every pair of distinct vertices is connected by a unique edge. In the context of planar graphs and graph coloring, K5 is significant because it serves as a classic example of a non-planar graph, meaning it cannot be drawn on a plane without edges crossing. Understanding K5 helps illustrate the limitations of planar graphs and plays a key role in concepts like graph coloring, where determining the minimum number of colors needed to color a graph's vertices without adjacent vertices sharing the same color becomes complex.

5 Must Know Facts For Your Next Test

1. K5 is a complete graph with 5 vertices, resulting in 10 edges because each vertex connects to every other vertex.
2. K5 is a crucial example in demonstrating Kuratowski's theorem, which states that a graph is non-planar if and only if it contains a subgraph that is homeomorphic to K5 or K3,3.
3. Any attempt to draw K5 on a flat surface will lead to edge crossings, confirming its status as a non-planar graph.
4. In graph coloring, K5 requires 5 different colors for proper coloring, as each vertex is adjacent to all other vertices.
5. The existence of K5 in a larger graph indicates that the larger graph cannot be planar due to its complete interconnectivity.

Review Questions

• How does K5 exemplify the concept of non-planarity in graphs?
  • K5 exemplifies non-planarity because it cannot be drawn in a plane without edges crossing. In a complete graph like K5, every vertex connects to every other vertex, creating an arrangement that inevitably results in edge intersections when attempting to represent it in two dimensions. This non-planarity serves as an essential case study in understanding the limitations and characteristics of planar graphs.
• Discuss how K5 impacts the understanding of graph coloring and its requirements.
  • K5 directly impacts the understanding of graph coloring by requiring five distinct colors for proper vertex coloring since every vertex connects to all other vertices. This necessitates using different colors for each vertex to ensure no adjacent vertices share the same color. The complexity introduced by K5 highlights how certain structures in graphs can drastically increase the number of colors needed for valid coloring solutions.
• Evaluate the significance of K5 in relation to Kuratowski's theorem and planar graph theory.
  • K5 holds significant importance in Kuratowski's theorem, which establishes criteria for identifying non-planar graphs. According to this theorem, if a graph contains either K5 or K3,3 as a subgraph, it is confirmed to be non-planar. This establishes K5 as a foundational example in planar graph theory, showing that certain arrangements within graphs inherently restrict their ability to exist without crossings on a plane, shaping our understanding of graph structures and properties.

Related terms

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

Graph Coloring: The assignment of labels or colors to the vertices of a graph such that no two adjacent vertices share the same color.

Euler's Formula: A formula that relates the number of vertices, edges, and faces of a connected planar graph, typically expressed as V - E + F = 2.