What is the minimum number of edges necessary in a simple planar graph with 15 regions?

In a simple planar graph, degree of each region is >= 3. So, we have 3 x |R| <= 2 x |E|. Thus, Minimum number of edges required in G = 23. Get more notes and other study material of Graph Theory.

What is the maximum number of edges possible in a simple planar graph with 4 vertices?

“If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces.” Euler’s Identity says, that for every planar graph of order n >= 3: the size m <= 3n – 6. That gives you an upper bound of 3*5-6 = 9 edges.

What is the maximum number of faces possible in a simple planar graph with 10 edges?

\(G\) has 10 edges, since \(10 = \frac{2+2+3+4+4+5}{2}\text{.}\) It could be planar, and then it would have 6 faces, using Euler’s formula: \(6-10+f = 2\) means \(f = 6\text{.}\)

How do you find the faces of a planar graph?

So, once we know that a graph G is planar, and then we may say that G has such and such number of faces without referring to any planar embedding. Theorem 1.8. 1: (Euler Formula) For a connected planar graph G = (V, E) with n vertices, m edges and f faces, n – m + f = 2.

Does a planar graph have to be connected?

Every maximal planar graph is a least 3-connected. If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces.

What is the smallest number of edges in a non planar graph?

10 edges, 5 vertices.

Can a simple graph be disconnected?

A simple graph, also called a strict graph (Tutte 1998, p. A simple graph may be either connected or disconnected. Unless stated otherwise, the unqualified term “graph” usually refers to a simple graph. A simple graph with multiple edges is sometimes called a multigraph (Skiena 1990, p.

How many vertices are there in a connected planar graph having 10 edges and 8 faces?

From eulerian formula : v+f−e=2: 10+8−e=2⟹e=16.

Are planar graphs connected?

How do you know if a graph is connected planar?

If a connected planar graph G has e edges and r regions, then r ≤ e. If a connected planar graph G has e edges, v vertices, and r regions, then v-e+r=2. If a connected planar graph G has e edges and v vertices, then 3v-e≥6. A complete graph K n is a planar if and only if n<5.

How to find the number of regions in a planar graph?

Let G be a connected planar simple graph with 20 vertices and degree of each vertex is 3. Find the number of regions in G. Thus, Total number of edges in G = 30. By Euler’s formula, we know r = e – v + 2. Thus, Total number of regions in G = 12. Let G be a connected planar simple graph with 35 regions, degree of each region is 6.

How to prove that complete bipartite graph is planar?

A complete bipartite graph K mn is planar if and only if m<3 or n>3. Example: Prove that complete graph K 4 is planar. Solution: The complete graph K 4 contains 4 vertices and 6 edges. We know that for a connected planar graph 3v-e≥6.Hence for K 4, we have 3×4-6=6 which satisfies the property (3). Thus K 4 is a planar graph.

What is the maximum number of colors in a planar graph?

Chromatic Number of any planar graph is always less than or equal to 4. Thus, any planar graph always requires maximum 4 colors for coloring its vertices.