Euler's formula
For a connected planar graph, the number of vertices plus the number of faces is two more than the number of edges. The face count includes the infinite outer face
\[v+f=e+2\]
- Check the conditions Use this identity only when the graph is connected and planar \(\displaystyle \text{connected and planar}\;\Rightarrow\;v+f=e+2\)
- Substitute known counts Replace the known vertex, edge or face counts in the identity \(\displaystyle v=6,\;e=9\;\Rightarrow\;6+f=9+2\)
- Solve the missing count Rearrange the equation after substitution \(\displaystyle 6+f=11\;\Rightarrow\;f=5\)
- Include the outer face The infinite region outside the drawing is one of the faces \(\displaystyle 4\text{ enclosed regions}+1\text{ outer region}=5\text{ faces}\)
Worked example
A connected planar graph has 7 vertices and 12 edges. Find its number of faces, including the outer face
\[v=7,\quad e=12,\quad 7+f=12+2\]
- \(\displaystyle v+f=e+2\)Use Euler's formula because the graph is connected and planar
- \(\displaystyle 7+f=12+2=14\)Substitute the 7 vertices and 12 edges
- \(\displaystyle f=14-7=7\)Subtract 7 from both sides to find the face count
- \(\displaystyle 7+7=12+2\)The result checks in Euler's formula, with the outer face included
Common mistakes
Misarranging Euler's formula
✕\(\displaystyle v+f=e-2\)✓\(\displaystyle v+f=e+2\)
The constant 2 is added to the edge count in Euler's formula for a connected planar graphOmitting the outer face
✕\(\displaystyle 4\text{ enclosed regions}\;\Rightarrow\;f=4\)✓\(\displaystyle 4\text{ enclosed regions}+1\text{ outer face}\;\Rightarrow\;f=5\)
The infinite outer region is included in the number of facesIgnoring connectedness
✕\(\displaystyle \text{disconnected planar graph}\;\Rightarrow\;v+f=e+2\)✓\(\displaystyle \text{connected planar graph}\;\Rightarrow\;v+f=e+2\)
This version of Euler's formula requires the planar graph to be connectedDifficulty levels
introFind the face count from small stated vertex and edge counts
standardRecord counts from a graph, verify Euler's formula and solve a second missing count
challengeUse a degree sum to find edges before applying Euler's formula and check the connectivity condition
examChoose the change or data set that is consistent with Euler's formula
Questions about this topic
Why is the outer face included?+
A planar drawing divides the whole plane into regions. The unbounded region outside the drawing is one of those regions, so it counts as a face
When can Euler's formula be used?+
Use v + f = e + 2 when the graph is connected and planar, with the face count including the outer face
Can Euler's formula find vertices or edges as well as faces?+
Yes. Rearrange v + f = e + 2 to make the unknown count the subject, provided the graph is connected and planar
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 8: Graphs and networks
In this chapter
Count the vertices, edges and loops of a graphDegrees, the degree sum, and even and odd verticesConnected graphs and bridgesIdentify isomorphic graphsPlanar graphs and drawings in planar formFaces and Euler's formula8C Euler's formula: A full network investigationClassify a route: walk, trail, path, circuit or cycleEulerian trails and circuits: existence and identification8E Eulerian trails and circuits: A full network investigationHamiltonian paths and cyclesRead a weighted graphThe shortest path by inspection8G Weighted graphs and shortest paths: A full network investigationTrees and spanning treesMinimum spanning trees by Prim's and Kruskal's algorithms8H Trees and minimum spanning trees: A full network investigation