Q 46,Paper 3,D 12 Two graphs A and B are shown below :

Which one of the following statement

is true ?

(A) Both A and B are planar.

(B) Neither A nor B is planar.

(C) A is planar and B is not.

(D) B is planar and A is not.

Answer (A)

Theorem 1. If v ≥ 3 then e ≤ 3v − 6;

Theorem 2. If v ≥ 3 and there are no cycles of length 3, then e ≤ 2v − 4.

Graph 1 has 4 v and 6 edges and 5faces.

Applying Theorem 1.

6<=12-6

Theorem 2.

4<=8-4.

Theorem 1 and 2 are satisfied .

Graph 2, has  8v and 12 edges.

Applying theorem 1.

12<=24-6.

Applying theorem 2.

12<=16-4.

Theorem 1 and theorem  2 is satisfied

Hence both are planar.