Question: Suppose that G = (V, E) is a loop-free planar graph with | V | = v, | E | = e, and k (G)

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) Prove that Corollary 11.3 remains valid if G is loop-free and planar but not connected.

Step by Step Solution

★★★★★

3.40 Rating (172 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a For each component C i V i E i 1 i n of G if e i E i and v i V i then ei vi 2 ri Su... 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 (8166).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

(a) Let k Z+, k ¥ 3. If G = (V, E) is a connected planar graph with |V| = v, |E| = e, and each cycle of length at least k, prove that (b) What is the minimal cycle length in K3,3? (c) Use parts...

(a) For n 1, let Pn-1 denote the path made up of n vertices and n - 1 edges. Let an be the number of independent subsets of vertices in Pn-1. (The empty subset is considered one of these independent...

If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a, b} of G either a or b is in K. The set K is a minimal covering if K - {x} fails to cover G for...

Suppose that G = (V, E) is a loop-free undirected graph. If G is 5-regular and |V| = 10, prove that G is nonplanar.

Let G = (V, E) be a loop-free undirected graph. (a) For each such graph, where |V| 3, find P(G, ) and show that in it the terms contain consecutive powers of . Also show that the coefficients of...

(a) Let G = (V, E) be a loop-free undirected graph, where |V| = 6 and deg(v) = 2 for all v V, Up to isomorphism how many such graphs G are there? (b) Answer part (a) for | V| -7. (c) Let G1 = (V1,...

Let G = (V, E) be a loop-free undirected graph. We call G color-critical if x(G) > x(G - v) for all v V. (a) Explain why cycles with an odd number of vertices are color-critical while cycles with an...

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...

(a) Let G = (V, E) be a loop-free undirected graph. Recall that G is called self-complementary if G and are isomorphic. If G is self-complementary (i) determine |E| if |V| = n; (ii) prove that G is...

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...

What is the difference between an analytical and a synthetic manufacturing process? Give an example of each type of process.

The following are survival times (in months) of a group of patients recorded in a clinical trial. 3 29+ 8+ 33 20+ 35 22+ 42+ 24 33 "*" indicates that the observation is right censored. (a) Find the...

Which of the following is the practice of raising small amounts of capital from a large number of people? shark funding venture capital float spreading crowdfunding individual funding

On January 1, 2004, Davis and Hershey owned Blackacre as tenants in common. On July 1, 2004, Davis made a written contract to sell Blackacre to Dibbert for $125,000. Pursuant to this contract,...

Let Q be an orthogonal matrix such that QA makes sense. Show that (QA)+ = A+QT.

Prove that for a diagonal matrix, the 1-, 2-, and - norms are the same.

Prove that for any square matrix A, ||A||22 ||A||1 ||A||

Consider the following annual returns of Molson Coors and International Paper: Compute each stock's average return, standard deviation, and coefficient of variation. (Round your answers to 2 decimal...

Consider the following two mutually exclusive projects: Year Cash Flow (X) Cash Flow (Y) 0 $15,300 $15,300 1 6,770 7,410 2 7,350 7,670 3 4,870 3,750 Requirement 1: (a) What is the IRR of Project X? (...

Find the present values of these ordinary annuities . Discounting occurs once a year. Do not round intermediate calculations. Round your answers to the nearest cent. $900 per year for 10 years at...