Euler's Formula

Euler's formula is a special relationship between the number of vertices ($V$), edges ($E$), and faces ($F$) in any connected planar graph. It states that:

$V - E + F = 2.$

This means that if you draw a connected graph on a plane without any edges crossing, count the number of vertices, edges, and regions (faces, including the outside), then this formula will always hold.

Important properties

• Applies to any connected planar graph (no edge crossings).

• Faces include the outer (infinite) region.

• If the graph is not connected, the formula generalizes to $V - E + F = 1 + c$, where $c$ is the number of connected components.

• Useful for checking if a graph can be drawn in the plane without crossings.

• Leads to important bounds on the number of edges in planar graphs.