Question
If a planar graph has the degree sequence {2;2;3;3;4;4;4}, how many faces will it have? Draw a planar graph with this degree sequence and number the faces to check your answer.
Answer :
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Skip in 10
To solve this problem, we will use Euler's formula for planar graphs: \[ V - E + F = 2 \] where: - \(V\) is the number of vertices, - \(E\) is the number of edges, - \(F\) is the number of faces. ### Step 1: Compute \(V\) and \(E\) The degree sequence given is: \[ \{2,2,3,3,4,4,4\} \] - The number of vertices \(V\) is 7 (since there are 7 degrees listed). - The sum of the degrees is: \[ 2 + 2 + 3 + 3 + 4 + 4 + 4 = 22 \] - By the handshaking lemma, the sum of all
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