2. Outline
• Definition
• Note
• Example
• Theorem (Euler’s Formula)
• Example
• Theorem
• Corollary
• Example
• Definition
• References
3. Definition
• A graph is planar if it can be drawn so that its edges do not cross.
4. Note
1. The Path Graph of order n is a planar graph for all 𝑛
2. The Cycle Graph of order n is a planar graph for all 𝑛
3. The wheel Graph of order n is a planar graph for all 𝑛
4. The star Graph of order n is a planar graph for all 𝑛
5. The complete Graph of order n is a planar graph for all 𝑛 ≤ 4
The complete bipartite graph is planar if either 𝑚 = 1 or 𝑛 = 1 also 𝐾2,2.
The graph 𝐾3,3 is a non-planar graph.
5. Definition
• Bounded regions defined by planar graphs are called faces (regions),
but unbound region called exterior face.
6. Example
This graph has a total of three faces: f = 3
Two interior faces (A, B)
One exterior face (C)
7. Theorem: Euler Formula
If 𝐺 is a connected plane graph, then
𝑉 − 𝐸 + 𝐹 = 2.
In the above theorem or formula, |𝑉|, |𝐸|, and |𝐹| denote the number
of vertices, edges, and faces of the graph G respectively.
8. A connected planar graph has 5 vertices and 7 edges, How many faces dose the
graph have?
Solution: Euler’s Formula
𝑉 − 𝐸 + 𝐹 = 2
5 − 7 + 𝐹 = 2
𝐹 = 2 + 2
𝐹 = 4
So number of faces is 4.
Example.
9. Theorem: For any (𝑝, 𝑞) planar graph in which every face is 𝐶𝑛
then, 𝑞 =
𝑛(𝑝−2)
n−2
Example: Does there exists a planar graph of order 18, and each
face is 𝐶5, if exists construct it, and then find the number of faces.
Solution: If the graph exists, then must satisfy the equation.
𝑞 =
5(18 − 2)
5 − 2
= 26.666
Which is not integer, therefore the graph does not exists.
10. Corollary
If G is any (𝑝, 𝑞) planar graph with 𝑝 ≥ 3, then 𝑞 ≤ 3𝑝 − 6, if G
has no triangle, then 𝑞 ≤ 2𝑝 – 4
11. Is K5 is a planar graph?
Solution: if K5 is planar graph, then must satisfies the inequality
𝑞 ≤ 3𝑝 − 6
𝑞 𝐾5 =
5(4)
2
= 10 𝑎𝑛𝑑 𝑝 = 5
∴ 10 ≤ 3 5 − 6 ⇒ 10 ≤ 15 − 6
10 ≤ 9
Sine 10 is not less or equal to 9
∴ 𝐾5 is not planar graph.
Example 1.
12. Is K3,3 is a planar graph?
Solution: is K3,3 is a planar graph then must satisfies the inequality
𝑞 ≤ 2𝑝 − 4
𝑞 𝐾3,3 = 3 × 3 = 9 𝑎𝑛𝑑 𝑝 = 3 + 3 = 6
∴ 9 ≤ 2 6 − 4 ⇒ 9 ≤ 12 − 4
9 ≤ 8
sine 9 not less or equal to 8
∴ 𝐾3,3 is not planar graph
Example 2.
13. Definition
A maximal planar graph is a graph in which no edge can be
added without losing planarity.
Example: the graphs 𝐾3,3 − 𝑒 and 𝐾5 − 𝑒 are maximal planar
graphs.
14. References
• Tutte, W. T. (1956). A theorem on planar graphs. Transactions of the
American Mathematical Society.
• Whitney, H. (1933). Planar graphs. In Hassler Whitney Collected
Papers. Boston, MA: Birkhäuser Boston.
• Biedl, T., Liotta, G., & Montecchiani, F. (2015). On visibility
representations of non-planar graphs.
