Question: Let G = (V, E) be an undirected connected loop-free graph. Suppose further that G is planar and determines 53 regions If, for some planar

Let G = (V, E) be an undirected connected loop-free graph. Suppose further that G is planar and determines 53 regions If, for some planar embedding of G, each region has at least five edges in its boundary, prove that | V | ≥ 82.

Step by Step Solution

★★★★★

3.33 Rating (153 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Proof Since each region has at le... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (8164).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 12.49. For n ¥ 0, let an count the number of spanning trees of G, whereas bn counts the number of these spanning trees that...

Let G = (V, E) be a loop-free undirected graph with |V| = n. Prove that G is a tree if and only if P(G, ) = ( - 1)n-1.

Let G = (V, E) be a loop-free connected undirected graph, where V = {a, b, c, . . . , h, i, j}. Ordering the vertices alphabetically, the depth-first spanning tree T for G - with a as the root-is...

Question 1 (Uniqueness of Planar Embeddings, 40 points). (a) Let G = (V, E) be a connected, planar graph with |E| = 3|V | 6. Show that any two planar embeddings of G have the same set of faces (in...

Subject - Graph Theory, Discrete Math (a) Let G = (V, E) be a connected planar graph such that |V | = 16 and every vertex has degree 4. It is embedded in the plane so that every region is either a...

(a) Let G = (V, E) be a loop-free weighted connected undirected graph. If e1 E with wt(ei)

nodes, but at least its bias can be quantified by Markov Chain L. INTRODUCTION analysis and thus can be corrected via appropriate re-weighting The popularity of online social networks (OSNs) in...

Suppose that G = (V, E) is a loop-free planar graph with | V | = v, | E | = e, and k (G) = the number of components of G. (a) State and prove an extension of Euler's Theorem for such a graph, (b)...

You$must$attach$this$cover$page$to$the$front$of$your$HW$or$you$will$lose$5$points.$ $ $ APMA$1860$Homework$2$ $ $ First$Name:$____________________________$$$Last$Name:$______________________________$...

4.10. A planar graph G is outerplanar if G can be drawn in the plane so that all of its vertices lie on the exterior boundary (on the unbounded region). (i) Show that K4 and K2,3 are not outerplanar....

How do characteristics such as substitutability and product life cycle stage influence the development of customer service goals and objectives?

= 3x + -6x - 5y = 0, -6x - 5y = 4, Round your answer to three decimal places. Find -6x - 5y dA, where R is the parallelogram enclosed by the lines 3x + 2y = 1, 3x + 2y = 9. This can be done directly...

1 1 - 5 . Under the hierarchy of GAAP, what sources of accounting principles have the most authority for state and local governments, federal government entities, and nongovemment entities? Where do...

Alda and Mattingly are residents of Unit I of Chimney Hills Subdivision. The lots owned by Alda and Mattingly are subject to the following restrictive covenant: Lots shall be for single-family...

Verify that when A has linearly independent columns, the definitions of pseudoinverse in this section and in Section 7.3 are the same.

Verify that the pseudoinverse (as defined in this section) satisfies the Penrose conditions for A (Theorem 7. 1 2 in Section 7.3).

Show that A+ is the only matrix that satisfies the Penrose conditions for A. To do this, assume that A' is a matrix satisfying the Penrose conditions: (a) AA'A = A, (b) A'AA ' = A', and (c) AA' and...

Refer to the stock options on Apple in the Figure 2.10. Suppose you buy an May expiration call option on 100 shares with the excise price of $145. 2-1. If the stock price in May is $153, will you...

2nd time posting this question. I'm having a hard time trying to get the commands to communicate with the classes and the floor. We have to have separate classes. I think i'm on the right path. I...

Frostbite Thermalwear has a zero coupon bond issue outstanding with a face value of $30,000 that matures in one year. The current market value of the firms assets is $36,400. The standard deviation...