Question: If G is a loop-free undirected graph with at least one edge, prove that G is bipartite if an only if x (G) = 2.

If G is a loop-free undirected graph with at least one edge, prove that G is bipartite if an only if x (G) = 2.

Step by Step Solution

★★★★★

3.48 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

If G V E is bipartite then F V 1 V 2 where V 1 V 2 0 and each edge is of the form xy whe... View full answer

blur-text-image
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)

Word file Icon

954-M-L-A-L-S (8222).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Prove Corollaries 11.4 and 11.5. Corollaries 11.4 Let G = (V, E) be a loop-free graph with n ( 2) vertices. If deg(u) (n - l)/2 for all v V, then G has a Hamilton path. Corollaries 11.5 If G = (V,...

Q:

Let G = (V, E) be a loop-free undirected graph with chromatic polynomial P(G, ) and |V| = n. Use Theorem 11.13 to prove that P(G, ) has degree n and leading coefficient 1 (that is, the coefficient of...

Q:

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.

Q:

8. Prove that any acyclic undirected graph with at least one edge must have a leaf (a vertex of degree exactly one)

Q:

Let G be a loop-free undirected graph, where = maxuV{deg(v)}. (a) Prove that x(G) + 1. (b) Find two types of graphs G, where x (G) = + 1.

Q:

A 9 - 2 . Matchings and covers Let G be a connected graph with at least one edge. ( a ) Let G be a graph that has a perfect matching ( note that there may be multiple perfect matchings in G . Prove...

Q:

I am still confused about the following two provements, can you help me out? 4. Let G be a connected graph with at least one edge. (a) {5 marks} Prove that each vertex of G is saturated by some...

Q:

Which of the following are always true: A weighted, undirected graph has at least one spanning tree. A directed graph cannot have exactly three vertices of odd out-degree. All forests can be colored...

Q:

5. Let G be a connected graph with at least one edge. (a) {5 marks} Prove that each vertex of G' is saturated by some maximum matching in G. (b) {4 marks} Provide a counterexample with explanations...

Q:

Let G be a connected, undirected graph with at least 3 vertices, and let G 3 be the graph obtained by connecting all pairs of vertices that are connected by a path in G of length at most 3. Prove...

Q:

Using AIRBNB I need THREE of the following: Financial Objectives, Measures and targets/timeline Customer Objectives, measures and targets/timeline Internal Business Process Objectives, measures and...

Q:

Briefly explain each filing status. How does filing status affect who must file income taxes? How does filing status affect the standard deduction amount?

Q:

4. A firms current profits are $900,000. These profits are expected to grow indefinitely at a constant annual rate of 2 percent. If the firms opportunity cost of funds is 4 percent, determine the...

Q:

Kraft is a limited partner of Johnson Enterprises, a lim ited partnership. As provided in the limited partner ship agreement, Kraft decided to leave the partnership and demanded that her capital...

Q:

Use statistics software to make a histogram of this data set: 44, 45, 51, 40, 28, 46, 34, 19. a. Based on the histogram, predict what a box plot of the data will look like. b. Use the software to...

Q:

Refer to the graph at right. a. Write a recursive formula for the sequence. What is the common difference? What is the value of u0? b. What is the slope of the line through the points? What is the...

Q:

Write the equation of the line that passes through the points of an arithmetic sequence with u0 = 7 and a common difference of 3.

Q:

For each research situation below, identify the independent and dependent variables. Classify each in terms of the level of measurement and whether the variable is discrete or continuous. (each...

Q:

7)A political campaign wants to rent photocopiers to make leaflets for a local election. There are two suitable machines: The ACTO machine costs $120 a month to rent, occupies 2.5 square meters of...

Q:

Youve collected the following information from your favorite financial website. 52-Week Price Stock (Dividend) Dividend Yield % PE Ratio Close Price Net Change Hi Lo 77.40 10.43 Acevedo .36 2.6 6...