Question: If G = (V, E) is an undirected graph, a spanning subgraph H of G in which each vertex has degree 1 is called a

If G = (V, E) is an undirected graph, a spanning subgraph H of G in which each vertex has degree 1 is called a one-factor (or perfect matching) for G.
a) If G has a one-factor, prove that |V| is even.
b) Does the Petersen graph have a one-factor? (The Petersen graph was first introduced in Example 11.19.)
c) In Fig. 13.31 we find the graph K4 in part (a), while part (b) provides the three possible one-factors for K4. How many one-factors are there for the graph K6?
d) For n ∈ Z+, let an count the number of one-factors that exist for the graph K2n. Find and solve a recurrence relation for an.

Step by Step Solution

★★★★★

3.53 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a A onefactor for a graph G VE consists of edges winch have no common v... 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 (8351).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

If G is an undirected graph with n vertices and e edges, let = minvV{deg(v)} and let = maxvV{deg(v)}. Prove that 2(e/n)

(a) If G = (V, E) is an undirected graph with |V| = v, |E| = e, and no loops, prove that 2e v2 - v. (b) State the corresponding inequality for the case when G is directed.

If G = (V, E) is an undirected graph with |V|-n and |E| = k, the following matrices are used to represent G. Let V = {v1, v2, . . . , Define the adjacency matrix A = (aI j)nÃn where aIJ = 1 if...

3 . H Another repurposing problem. Let G = ( V , E ) be an undirected graph with vertices V = { 0 , 1 , 2 , 3 , , n 1 } , and edges defined by an array of adjacency lists Adj [ 0 . . n 1 ] . We say...

4. We need two quick definitions before the following question: Definition 0.3. Let G = (V, E) be an undirected graph, and let v EV. The degree of v is the number of edges in G involving v....

4. (Independence number) Let G (V, E) be an undirected graph with n vertices. A subset SCV is independent if no pair of vertices in S is adjacent. In other words, S is independent in G exactly if S...

If G = (V, E) is an undirected graph, a subset I of V is called independent if no two vertices in I are adjacent. An independent set I is called maximal if no vertex v can be added to I with I ª...

Find the minimum cost let G= {V E be an undirected connected graph. A subgraph + = {VE3 of G is a spanning tree iff & is a tree. By We have 2 methods to generate min. cost spanning treet 1) Kruskal...

Could someone help me with the problems below Let d be the maximum degree of the vertices in a graph G. Prove that we can color G with d + 1 colors. Consider this statement: In any directed graph G =...

Could someone please help me with the problems below Could someone help me with the problems below Let d be the maximum degree of the vertices in a graph G. Prove that we can color G with d + 1...

An annual report for Ford Motor Company contained the following note: Significant Accounting Policies Depreciation of Property, Plant, and Equipment. Property and equipment are stated at cost and...

Identify four indirect costs you might find on a moderately complex project. Why are these costs classified as indirect?

"Bulldog" bonds are Question 1 0 options: a ) Yen - denominated foreign bonds originally sold in Japan b ) Dollar - denominated foreign bonds originally sold to U . S . investors c ) Pound sterling -...

The county intends to rezone an area from industrial use to residential use. Land within the affected area is largely undeveloped, but nonetheless it is expected that the proposed action will reduce...

Solve. a. log 35 + log 7 = log x b. log 500 - log 25 = log x c. log (1/8 = x log8 d. 15(9.4)x = 37000 e. f. log6 342 = 2x

Suppose the signal strength in a fiber-optic cable diminishes by 15% every 10 mi. a. What percentage of the original signal strength is left after a stretch of 10 mi? b. Create a table of the...

The graph of an exponential function passes through the points (4, 50) and (6, 25.92). a. Find the equation of the exponential function. b. What is the rate of change? Does the equation model growth...

Discuss the I 2 C communication process along with the two I 2 C mbed ports.

Crosby Industries has a debt-equity ratio of 1.1. Its WACC is 13 percent, and its cost of debt is 4 percent. There is no corporate tax. Requirement 1: What is Crosby's cost of equity capital? (Do not...

In the 2004 presidential election, 18.3 million 25-34 year-olds reported that they voted. If this represents 46.9% of the entire 25-34 year-old population, how many 25-34 year-olds were there in 2004...