Question

To Determine Into how many regions is the plane divided by a planar representation of this graph?

Short answer

There are 6 regions.

Step-by-step solution

 Given

A connected planar graph has 8 vertices, each of degree 3.

\(v = 8\)

\(\deg \left( {{v_i}} \right) = 3\)

The Concept ofEuler's formula

It is written r + v = e + 2, where r is the number of faces, v the number of vertices, and e the number of edges.

Determine the edges

A connected planar graph has 8 vertices, each of degree 3.

\(v = 8\)

\(\deg \left( {{v_i}} \right) = 3\)

Since the degree of each vertex is 4, there are 4 edges connecting to each of the 6 vertices and thus there are\(3 \cdot 8 = 24\)connections to all vertices.

Since each edge has 2 connections to a vertex, the graph has\(\frac{{24}}{2} = 12\)edges.

\(e = \frac{{3 \cdot 8}}{2} = \frac{{24}}{2} = 12\)

By Euler's formula,

\(r = e - v + 2 = 12 - 8 + 2 = 6\)