Planar graph

Definition

A planar graph is a type of graph that can be drawn on a two-dimensional plane without any edges crossing each other. This property allows planar graphs to represent relationships in a way that makes them easier to visualize and analyze. Planar graphs are fundamental in various fields, including topology and graph theory, particularly when studying concepts like the four-color theorem.

5 Must Know Facts For Your Next Test

1. Planar graphs can have at most 3V - 6 edges, where V is the number of vertices, for graphs with V ≥ 3.
2. The four-color theorem states that any planar graph can be colored with no more than four colors such that no adjacent regions share the same color.
3. Planar graphs are significant in map coloring problems, where regions on a map are represented as vertices and shared borders as edges.
4. Not all graphs are planar; for example, K5 (complete graph on five vertices) is non-planar as it cannot be drawn without edge crossings.
5. Algorithms exist to determine if a given graph is planar, such as using Kuratowski's theorem or implementing planarity testing algorithms.

Review Questions

• How do planar graphs relate to Euler's formula and what significance does this relationship hold in understanding their structure?
  • Planar graphs are directly related to Euler's formula, which states that for a connected planar graph, the relationship V - E + F = 2 holds true. Here, V represents the number of vertices, E represents the number of edges, and F represents the number of faces. This relationship helps in understanding the limitations on how many edges can exist in relation to vertices and faces, providing insight into the structure and properties of planar graphs.
• Discuss how Kuratowski's theorem aids in identifying non-planar graphs and its implications for graph theory.
  • Kuratowski's theorem is a key tool in graph theory that helps identify non-planar graphs by stating that a finite graph is non-planar if it contains a subdivision of either K5 or K3,3. This theorem simplifies the process of determining planarity by providing concrete criteria to check against. The implications extend beyond just planarity; they influence areas like topological embeddings and help understand more complex relationships in graph structures.
• Evaluate the impact of the four-color theorem on applications involving planar graphs and its significance in real-world scenarios.
  • The four-color theorem has a profound impact on various applications involving planar graphs, particularly in fields such as cartography and scheduling. It ensures that any map can be colored using just four colors without adjacent regions sharing the same color, which simplifies many practical problems. Its significance also lies in its computer-assisted proof, which showcases how computational methods can resolve complex mathematical questions, paving the way for future explorations in both mathematics and computer science.

Related terms

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

Kuratowski's theorem: A theorem stating that a finite graph is planar if and only if it does not contain a subdivision of the complete graph K5 or the complete bipartite graph K3,3.

graph coloring: An assignment of labels (colors) to the vertices of a graph so that no two adjacent vertices share the same color, which is crucial for solving problems like the four-color theorem.