If G is a simple connected 3-regular planar graph where every region is bounded by exactly 3 edges, then the edges of G is
A)3
B)4
C)6
D)5

Question

Quizplus · Accepted Answer

A 3-regular graph means every vertex has a degree of 3. Since the graph is planar and every region is bounded by 3 edges, we can use Euler's formula for planar graphs, $V - E + F = 2$, where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces (regions). Each edge contributes to 2 regions since the graph is simple and planar. Therefore, $3F = 2E$, because each face is bounded by 3 edges but each edge is counted twice, once for each of its two adjacent faces. Substituting $F = \frac{2E}{3}$ into Euler's formula gives $V - E + \frac{2E}{3} = 2$, simplifying to $V = \frac{E}{3} + 2$. Since the graph is 3-regular, $3V = 2E$. Substituting $V = \frac{E}{3} + 2$ into $3V = 2E$ gives $E = 6$.

If G Is a Simple Connected 3-Regular Planar Graph Where