Planar Graphs and Euler's formula
A planar graph can be drawn with no edges crossing, and for any connected planar graph Euler's formula links the vertices, edges and faces: \(v - e + f = 2\). This NSW Year 9 Advanced maths subtopic applies the formula (NESA outcome MA5‑NET‑P‑01).
You will count vertices, edges and faces, including the outside face, redraw graphs in planar form, and use Euler's formula to find a missing value or spot a counting error. You will apply it to a pyramid, a solid with \(12\) faces, a map of walking tracks and a soccer ball, and use it to show that the complete graph \(K_5\) cannot be drawn without crossings. It is part of Year 9 Advanced Properties of Geometric Figures.
Theory
A planar graph can be drawn with no two edges crossing, and its edges then split the page into faces. For every connected planar graph, Euler's formula \(v - e + f = 2\) links the vertices, edges and faces. This Year 9 Advanced maths page shows how to count faces, redraw a graph in planar form, and use the formula for solids, degrees and face counts.
A graph is planar if it can be drawn in the plane so that no two edges cross. A drawing with no crossings is in planar form. A drawing with crossings does not prove a graph is non-planar: it may be possible to redraw it. A non-planar graph is one that can never be drawn without a crossing.
A planar drawing splits the page into regions called faces. The unbounded region all around the outside of the graph is also a face, and it is the one most often forgotten.
For a connected planar graph with \(v\) vertices, \(e\) edges and \(f\) faces, Euler's formula says \(v - e + f = 2\). One way to see it: start from a single vertex (\(1 - 0 + 1 = 2\)). Adding an edge to a new vertex adds \(1\) to both \(v\) and \(e\). Adding an edge between two existing vertices closes a loop, adding \(1\) to both \(e\) and \(f\). Either way \(v - e + f\) stays \(2\).
The corners and edges of a solid such as a prism or a pyramid can be flattened into a planar graph without changing the counts, so Euler's formula holds for those solids too (with \(v\) = corners and \(f\) = faces).
For a connected planar graph:
Rearranged for each letter:
How to count faces and use Euler's formula
- Check the drawing is in planar form. If two edges cross, redraw it first: move a vertex, or take an edge round the outside.
- Count \(v\) and \(e\) from a list, ticking each one off. Count the faces as enclosed regions, then add \(1\) for the outside.
- Check with \(v - e + f = 2\). If it does not give \(2\), a count is wrong, and the usual culprit is the outside face.
- Solve for a missing count by substituting the two you know into the rearranged formula.
Harder questions
- Testing a claim: substitute the three numbers. If \(v - e + f \ne 2\), no connected planar graph has them.
- Two unknowns linked by a sentence (for example "twice as many edges as vertices"): write one in terms of the other, substitute and solve the linear equation.
- Expressions: if \(v\), \(e\) and \(f\) are given in terms of \(x\), substitute all three, collect like terms and solve.
- Adding to a graph: recount \(v\) and \(e\) and use the formula again. An edge that closes a loop adds a face.
Vertices: \(A\) to \(G\), so \(v = 7\). Edges: the six sides of the hexagon, \(BF\), \(CE\) and \(DG\), so \(e = 9\).
Faces: \(ABF\), \(BCEF\), \(CDE\) and the outside, so \(f = 4\). The edge \(DG\) does not enclose a region, so it adds no face.
| \(v - e + f\) | \(=\) | \(7 - 9 + 4\) |
| \(\) | \(=\) | \(2 \ \checkmark\) |
The corners and edges of a solid can be flattened into a connected planar graph, so \(v - e + f = 2\). Make \(e\) the subject.
| \(e\) | \(=\) | \(v + f - 2\) |
| \(\) | \(=\) | \(10 + 7 - 2\) |
| \(\) | \(=\) | \(\mathbf{15}\) |
A pentagonal prism fits: \(2\) pentagons and \(5\) rectangles make \(7\) faces, with \(10\) corners and \(5 + 5 + 5 = 15\) edges.
Euler's formula needs \(e\). Get it from the degrees with the handshake rule.
| \(2e\) | \(=\) | \(8 \times 3 = 24\) |
| \(e\) | \(=\) | \(\mathbf{12}\) |
| \(8 - 12 + f\) | \(=\) | \(2\) |
| \(f\) | \(=\) | \(\mathbf{6}\) |
These are the numbers for a cube: \(8\) corners, \(12\) edges and \(6\) faces. The drawing shows the cube flattened, with the outside region as the sixth face.
Count the sides of the faces two ways. Face by face, there are \(4f\) edge-sides. Edge by edge, each edge has a face on each side, so there are \(2e\).
| \(4f\) | \(=\) | \(2e\) |
| \(e\) | \(=\) | \(2f\) |
Substitute into Euler's formula with \(v = 10\):
| \(10 - 2f + f\) | \(=\) | \(2\) |
| \(10 - f\) | \(=\) | \(2\) |
| \(f\) | \(=\) | \(\mathbf{8}\) |
| \(e\) | \(=\) | \(2 \times 8 = \mathbf{16}\) |
Check: \(10 - 16 + 8 = 2\), and \(4 \times 8 = 32 = 2 \times 16\).
Common pitfalls
Frequently asked questions
What is Euler's formula for planar graphs?
For any connected planar graph, the number of vertices minus the number of edges plus the number of faces equals \(2\): \(v - e + f = 2\). The outside region counts as a face.
What is a planar graph?
A planar graph is a graph that can be drawn on a flat page with no two edges crossing. A particular drawing may have crossings, but if any crossing-free drawing exists, the graph is planar.
Does the outside count as a face?
Yes. The unbounded region around the outside of a planar drawing is a face, so a single triangle has 2 faces: the inside and the outside.
Does Euler's formula work for 3D shapes?
Yes, for solids such as prisms and pyramids. Their corners, edges and faces can be flattened into a connected planar graph, so corners \(-\) edges \(+\) faces \(= 2\). A cube has \(8 - 12 + 6 = 2\).
Is every graph planar?
No. Some graphs can never be drawn without a crossing. Two famous examples are the complete graph on 5 vertices and the graph joining each of 3 houses to each of 3 utilities.