GATE | GATE-CS-2014-(Set-3) | Question 65

Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d ≥ 3, which one of the following is TRUE?
(A) In any planar embedding, the number of faces is at least n/2 + 2
(B) In any planar embedding, the number of faces is less than n/2 + 2
(C) There is a planar embedding in which the number of faces is less than n/2 + 2
(D) There is a planar embedding in which the number of faces is at most n/(d+1)


Answer: (A)

Explanation:

Euler's formula for planar graphs:

v − e + f = 2.

v → Number of vertices
e → Number of edges
f → Number of faces

Since degree of every vertex is at least 3, 
below is true from handshaking lemma (Sum of
degrees is twice the number of edges)

3v ≤ 2e
3v/2 ≤ e 
Putting these values in Euler's formula. 
v - 3v/2 + f ≥ 2 
f ≥ v/2 + 2

 

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