Planar Graphs

Drawing without crossings

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A planar graph can be drawn without crossings. Its drawing has faces, including the outside.
• Euler’s formula: V − E + F = 2.
• A planar graph with V ≥ 3 has at most 3V − 6 edges, so K₅ is not planar.
• K₃,₃ (the utilities puzzle) is not planar either. These two are the only obstacles.

Steps

1. Untangle
2. Faces and Euler’s formula
3. Count the faces
4. A bigger tangle
5. Three utilities
6. How many lines fit?
7. Always twelve