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Suppose that a connected planar graph has six vertices, each of degree four. Into how many regions is the plane divided by a planar representation of this graph?

1. $6$
2. $8$
3. $12$
4. $20$

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+1

formulas used

Sum of degree$=2e$

by formula $r=e-v+2$
0
Ans is 8

Method 1 (brute force) :-

Method 2 :-

We know that,

sum of degree of vertices = 2 * number of edges (handshaking theorem)

$\implies 4+4+4+4+4+4 = 2 * e$

$\implies e = 12$

Also according to Euler's formula,

$r=e-v+2$

$\implies r= 12 -6 +2$

$\implies r= 12 -6 +2$

$\implies r= 8$

$\therefore$ Option $2.$ is correct.

by Boss (17.2k points)
edited

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