General Mathematics · Unit 4 · Graphs and networks · Planar graphs, paths and cycles

Apply Euler’s formula to solve problems relating to planar graphs.  𝑣 + 𝑓 − 𝑒 = 2 where 𝑣 is number of vertices, 𝑓 is number of faces and 𝑒 is number of edges

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Question 1

A planar graph has 7 vertices and 12 edges. (a) Use Euler's formula to find the number of faces in this graph. (b) Verify that your answer is correct by showing the application of the formula.

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Question 2

A planar graph has 7 edges and 4 faces. Apply Euler's formula to find the number of vertices.

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Question 3

A planar graph has 8 edges and 5 faces. (a) Use Euler's formula to find the number of vertices. (b) Verify your answer by checking that the formula \ \( v + f - e = 2 \) holds.

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Question 4

A planar graph has \(8\) vertices and \(5\) faces. (a) Use Euler's formula to determine the number of edges in this graph. (2 marks) (b) A new edge is added to the graph, connecting two existing vertices. Calculate the new number of faces, showing your working. (2 marks)

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Question 5

A planar graph has 8 vertices and 12 edges. Using Euler's formula $v + f - e = 2$, how many faces does the graph have?

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Solve practical problems involving semi-Eulerian graphs and Eulerian graphs.

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