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Previous Year Question (PYQs)

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?




Solution

Given,

$V=6$

Each vertex has degree $4$.

So, sum of degrees is

$6 \times 4=24$

By handshaking theorem,

$2E=24$

$E=12$

For a connected planar graph, Euler’s formula is

$V-E+R=2$

Substitute $V=6$ and $E=12$.

$6-12+R=2$

$R=8$


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