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Faces and Euler's formula

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Which method is valid for a connected planar graph?

Year 12, VCE Units 3 & 4, Chapter 13, 4 difficulty levels, No calculatorBuild a worksheet

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

\[v+f=e+2\]
  1. Check the conditions Use this identity only when the graph is connected and planar \(\displaystyle \text{connected and planar}\;\Rightarrow\;v+f=e+2\)
  2. Substitute known counts Replace the known vertex, edge or face counts in the identity \(\displaystyle v=6,\;e=9\;\Rightarrow\;6+f=9+2\)
  3. Solve the missing count Rearrange the equation after substitution \(\displaystyle 6+f=11\;\Rightarrow\;f=5\)
  4. 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

\[v=7,\quad e=12,\quad 7+f=12+2\]
  1. \(\displaystyle v+f=e+2\)Use Euler's formula because the graph is connected and planar
  2. \(\displaystyle 7+f=12+2=14\)Substitute the 7 vertices and 12 edges
  3. \(\displaystyle f=14-7=7\)Subtract 7 from both sides to find the face count
  4. \(\displaystyle 7+7=12+2\)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 graph
Omitting 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 faces
Ignoring 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 connected

Difficulty 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 Euler's formula when the graph is connected and planar, counting the outer face

Can Euler's formula find vertices or edges as well as faces?+

Yes. Rearrange Euler's formula 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 12, VCE Units 3 & 4, General Mathematics, chapter 13: Graphs and networks

In this chapter

Count the vertices, edges and loops of a graphDegrees, the degree sum, and even and odd verticesGraph types, subgraphs, connection models and method selectionConnected graphs and bridgesIdentify isomorphic graphsPlanar graphs and drawings in planar formFaces and Euler's formulaRepresent a network in a matrixIdentify the network a connection matrix describesClassify a route: walk, trail, path, circuit or cycleEulerian trails and circuits: existence and identificationHamiltonian paths and cyclesRead a weighted graphThe shortest path by inspectionDijkstra's algorithm for shortest distances and routesTrees and spanning treesMinimum spanning trees by Prim's and Kruskal's algorithmsA full network investigation