Euler formula for planar graphs
Relate vertices, edges, and faces in a connected planar embedding.
Euler formula for planar graphs is one of 4 graphs formulas in the discrete math section of this library, and it is used at university level.
Why euler formula for planar graphs works
Start from a tree drawn on the page. It has one face, the outside, and one less edge than it has vertices, so the alternating total works out to 2. Now add edges one at a time: each new edge closes a loop and slices an existing face into two, so the edge count and the face count both rise by 1 and cancel. The total never moves off 2.
What each symbol means
$F$ includes the unbounded outer face.
Euler formula for planar graphs: when it holds
The graph must be connected and planar; for $c$ components use $|V|-|E|+|F|=1+c$.
When it stops applying
It needs the graph to be connected and drawn without crossings. Two separate triangles give 6 − 6 + 3 = 3, matching the version 1 + c for c pieces, and a graph like K₅ cannot be drawn flat at all, so it has no faces to count.
Euler formula for planar graphs: a worked example
A triangle has $3-3+2=2$.
The mistake to avoid
What people do: Students count only the enclosed regions and leave out the unbounded area around the drawing.
Why it goes wrong: The outside counts as a face too. Skipping it makes every answer come out as 1 instead of 2, which looks like the formula is broken.
Do this instead: Include the outer region in F. A triangle has 3 vertices, 3 edges, and 2 faces, the inside and the outside, so 3 − 3 + 2 = 2 as promised.
Euler formula for planar graphs: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $F$ includes the unbounded outer face.
- Check the conditions before substituting. The graph must be connected and planar; for $c$ components use $|V|-|E|+|F|=1+c$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most discrete math slips.
Where this formula fits
- Subject
- Discrete Math formulas — 11 entries in this library
- Topic
- Graphs
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where euler formula for planar graphs comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Discrete Math Calculator — check your substitution and the value it produces.
- Study discrete math — the subject guide that explains the ideas these formulas compress.
- Discrete Math Practice — questions that make you retrieve the formula instead of recognising it.
- All 11 discrete math formulas — the full grouped reference, or the complete formula library.
Questions about euler formula for planar graphs
Does the outside really count as a face?
Yes, and there is a reason. Drawing the graph on a sphere instead of a page turns that outer region into an ordinary patch, and every face looks the same, which is the natural home of this formula.
What if the graph comes in separate pieces?
Use |V| − |E| + |F| = 1 + c, where c counts the pieces. Each extra component adds 1 to the total, since the pieces share one outer face but bring their own vertices and edges.
Why can K₅ not be drawn without crossings?
Because a simple planar graph must satisfy |E| ≤ 3|V| − 6, which comes from Euler's formula. With 5 vertices that caps the edges at 9, but K₅ needs 10.
Does this work for solids like a cube?
Yes, and that is the original setting. A cube has 8 corners, 12 edges, and 6 faces, and 8 − 12 + 6 = 2, because the surface of any convex solid can be flattened into a planar drawing.