Euler's Formula

Graph Theory · Axiom Academy

Discover the beautiful relationship between vertices, edges, and faces in planar graphs 1. Counting Vertices, Edges, and Faces Let's start by understanding what we're counting in a planar graph. A planar graph can be drawn on a plane without any edges crossing. Watch as we build a simple planar graph and count each component: 2. Euler's Formula: V - E + F = 2 For any connected planar graph , the number of vertices, edges, and faces always satisfy this remarkable equation. Let's verify it with a more complex example: 3. Proof Sketch by Induction on Edges We can prove Euler's formula by adding edges one at a time and showing that V - E + F remains constant at 2. Base Case: Start with a single vertex and no edges. Then V = 1, E = 0, F = 1, so V - E + F = 1 - 0 + 1 = 2. Inductive Step: When we add an edge, one of two things happens: Since the formula holds for the base case and is preserved when adding any edge, it holds for all connected planar graphs! 4. Applications and Consequences Euler's formula has many powerful applications in graph theory and beyond: 5. Generalizing to Disconnected Graphs What happens when a planar graph is not connected? Euler's formula generalizes beautifully to handle multiple components.

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