Question: (a) Let G = (V,E) be a loop-free connected graph with | V | 11. Prove that either G or its complement must be

(a) Let G = (V,E) be a loop-free connected graph with | V | ≥ 11. Prove that either G or its complement must be nonplanar.
(b) The result in part (a) is actually true for | V | ≥ 9, but the proof for | V | = 9, 10, is much harder. Find a counterexample to part (a) for | V | = 8.

Step by Step Solution

★★★★★

3.33 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Suppose that G VE with V 11 Then VE 1 where a b E 1 iff a b g Let e E e 1 E 1 If ... 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 (8168).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let G = (V, E) be a loop-free connected undirected graph with |V| 2. Prove that G contains two vertices v, w, where deg (v) = deg(w).

(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 connected planar graph. If G is isomorphic to its dual and |V| = n, what is |E|?

Let G = (V, E) be a loop-free connected graph with | V | = v. If | E | > (v/2)2, prove that G cannot be bipartite.

Let G = (V, E) be a loop-free connected undirected graph. Let H be a subgraph of G. The complement of H in G is the subgraph of G made up of those edges in G that are not in H (along with the...

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...

Question 3. (1 marks) Let G = (V, E) be an undirected connected graph with n nodes and m edges. In this question you are asked to orient the graph, i.e. pick one direction for each edge: an edge (u....

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...

Let G = (V, E) be a loop-free connected undirected graph, and let {a, b} be an edge of G. Prove that {a, b} is part of a cycle if and only if its removal (the vertices a and b are left) does not...

Let G = (V, E) be a loop-free connected undirected graph with v V. (b) If v is an articulation point of G, prove that v cannot be an articulation point of . G- v = G - v.

Twenty two (22) grams of unknown solute was mixed in 100g of ethanol. The boiling point of ethanol increased by .30C. What is the molar mass of the unknown solute?

The mayor of Springfield justifies a proposed city bus system by pointing out it has a benefit-cost ratio of 3 to2. What does this mean?

E3.9 (LO 2, 3, 4) (Income Statement with Retained Earnings) Presented below is information related to Tao Ltd. for the year 2025 (amounts in thousands). Net sales HK$1,200,000 Write-off of inventory...

Robert and Stanley held legal title of record to adjacent tracts of land, each consisting of a number of five acres. Stanley fenced his five acres in 1985, placing his east fence fifteen feet onto...

In Exercises 1-2, find the best quadratic approximation to f on the interval [0, l]. 1. f(x) = x3 2. f(x) = x 3. f(x) = ex

Show that 1 is orthogonal to cos kx and sin kx in [- , ] for k 1.

Show that cos jx is orthogonal to cos kx in [- n, n] for j k, j, k 1.

Leah, Inc., is proposing a rights offering. Presently there are 400,000 shares outstanding at $44 each. There will be 100,000 new shares offered at $35 each. a. What is the new market value of the...

For each of the following queries: a- Draw the query graph. b- Draw the query tree corresponding to the relational algebra expression. c- Draw the initial query tree, then show how the same query...

Question Emma, a golfer, claims that her drive distance is less than175 meters, on average. Several of her friends do not believe her, so she decides to do a hypothesis test, at a1% significance...