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🧮combinatorics review

Planar graph

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A planar graph is a graph that can be drawn on a flat surface without any edges crossing each other. This concept is crucial in understanding the relationships between different vertices and edges, as it allows for a clearer visualization of how these elements interact. Planar graphs are closely connected to various properties and theorems, including those related to coloring and network design.

5 Must Know Facts For Your Next Test

  1. A graph is planar if it can be embedded in the plane without edge crossings, allowing for a clear representation of relationships.
  2. The maximum number of edges in a simple planar graph with V vertices is 3V - 6, as long as V is greater than or equal to 3.
  3. Planar graphs can have dual graphs, where the vertices correspond to the faces of the original graph and vice versa.
  4. The Four Color Theorem states that four colors are sufficient to color any planar graph so that no two adjacent vertices share the same color.
  5. If a graph contains a subgraph that is homeomorphic to K5 or K3,3, it cannot be drawn as a planar graph.

Review Questions

  • How does Euler's Formula apply to planar graphs, and what does it reveal about their structure?
    • Euler's Formula states that for any connected planar graph, the relationship between the number of vertices (V), edges (E), and faces (F) can be expressed as V - E + F = 2. This formula reveals that there are inherent structural limitations in how planar graphs can be formed. By analyzing this relationship, we can understand how adding or removing vertices and edges affects the overall structure of the graph, including its connectivity and complexity.
  • Discuss Kuratowski's Theorem and its importance in determining whether a graph is planar.
    • Kuratowski's Theorem provides a definitive test for planarity by stating that a finite graph is planar if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph). This theorem is significant because it gives us a method for assessing planarity without needing to attempt to draw the graph physically. By identifying these non-planar structures within a graph, we can efficiently determine its planarity.
  • Evaluate how the Four Color Theorem impacts practical applications in various fields involving planar graphs.
    • The Four Color Theorem asserts that four colors are sufficient to color any planar graph so that adjacent vertices do not share the same color. This theorem has profound implications across multiple disciplines, including map coloring, scheduling problems, and network design. In practical terms, it ensures that resources can be allocated efficiently without conflicts. For example, when designing communication networks or traffic systems represented as graphs, applying this theorem helps minimize interference and optimize connectivity.