Question

Suppose G is a simple, nonconnected graph with n vertices that is maximal with respect to...

Suppose G is a simple, nonconnected graph with n vertices that is maximal with respect to these properties. That is, if you tried to make a larger graph in which G is a subgraph, this larger graph will lose at least one of the properties (a) simple, (b) nonconnected, or (c) has n vertices.

What does being maximal with respect to these properties imply about G?G? That is, what further properties must GG possess because of this assumption?

In this activity you hopefully decided that a graph as described in the activity must:

  1. have exactly two connected components; and
  2. each connected component must be a complete graph.

Prove these two statements.

Notes:

  • Write a separate proof for each of the two statements.
  • Your proofs should be general proofs; they should not rely on examples.
  • Be aware that there isn't just one unique graph G as described in the activity for each value of n. In fact, for each n≥4 there will be n/2 different versions of such maximal simple nonconnected graphs (rounding n/2 down in case n is odd), and the different versions will have different numbers of edges while still having the same number of vertices.

Hint: For each of the two proofs, you might consider arguing by contrapositive or by contradiction.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let n be a positive integer and G a simple graph of 4n vertices, each with...
Let n be a positive integer and G a simple graph of 4n vertices, each with degree 2n. Show that G has an Euler circuit. (Hint: Show that G is connected by assuming otherwise and look at a small connected component to derive a contradiction.)
I.15: If G is a simple graph with at least two vertices, prove that G has...
I.15: If G is a simple graph with at least two vertices, prove that G has two vertices of the same degree.    Hint: Let G have n vertices. What are possible different degree values? Different values if G is connected?
Let G be a connected simple graph with n vertices and m edges. Prove that G...
Let G be a connected simple graph with n vertices and m edges. Prove that G contains at least m−n+ 1 different subgraphs which are polygons (=circuits). Note: Different polygons can have edges in common. For instance, a square with a diagonal edge has three different polygons (the square and two different triangles) even though every pair of polygons have at least one edge in common.
Let G be an undirected graph with n vertices and m edges. Use a contradiction argument...
Let G be an undirected graph with n vertices and m edges. Use a contradiction argument to prove that if m<n−1, then G is not connected
Suppose we are going to color the vertices of a connected planar simple graph such that...
Suppose we are going to color the vertices of a connected planar simple graph such that no two adjacent vertices are with the same color. (a) Prove that if G is a connected planar simple graph, then G has a vertex of degree at most five. (b) Prove that every connected planar simple graph can be colored using six or fewer colors.
Graph Theory. A simple graph G with 7 vertices and 10 edges has the following properties:...
Graph Theory. A simple graph G with 7 vertices and 10 edges has the following properties: G has six vertices of degree a and one vertex of degree b. Find a and b, and draw the graph. Show all work.
Let G = (V,E) be a graph with n vertices and e edges. Show that the...
Let G = (V,E) be a graph with n vertices and e edges. Show that the following statements are equivalent: 1. G is a tree 2. G is connected and n = e + 1 3. G has no cycles and n = e + 1 4. If u and v are vertices in G, then there exists a unique path connecting u and v.
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff...
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff for every set of 3 distinct vertices, a, b and c, there is an a,c-path that contains b.
Let G be a simple graph with at least two vertices. Prove that there are two...
Let G be a simple graph with at least two vertices. Prove that there are two distinct vertices x, y of G such that deg(x)= deg(y).
Show that if G is a graph with n ≥ 2 vertices then G has two...
Show that if G is a graph with n ≥ 2 vertices then G has two vertices with the same degree.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT