GATE CS 2014 Set 3 — Question 62
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Engineering Mathematics → Graph Theory (Math) → Planarity
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Question
Let denote the minimum degree of a vertex in a graph. For all planar graphs on vertices with , which one of the following is TRUE?
Correct answer
(A) In any planar embedding, the number of faces is at least (n)/(2) + 2
Solution
For a connected planar graph, Euler's formula states , where . Thus, .The sum of degrees is . Since the minimum degree is , we have:Given , we have , or .Substituting this into the expression for :Thus, . This inequality holds for any planar embedding of such a graph.If the graph is not connected, let be the number of connected components. Euler's formula is . Then . Since , , and the same bound applies.
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