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Chapter 11: Problem 14

a) Find a graph \(G\) where both \(G\) and \(\bar{G}\) are connected. b) If \(G\) is a graph on \(n\) vertices, for \(n \geq 2\), and \(G\) is not connected, prove that \(\bar{G}\) is connected.

Short Answer

Expert verified
a) The graph \(G\) consists of three vertices all connected, hence is connected. The complement \(\bar{G}\) contains no edges, hence is also connected. b) Proof: \(G\) is not connected implies that there exist at least two vertices for which there is no path in \(G\). However, these vertices are now connected in \(\bar{G}\), ensuring a path exists between every pair of vertices. Thus, if \(G\) is not connected, then \(\bar{G}\) is connected.

Step by step solution

01

Part A: Find a Graph

For this problem, a simple graph needs to be found. The graph should consist of three vertices all connected to each other, labeled as such: A-B, B-C, A-C. This represents a graph \(G\) which is connected.
02

Part A: Find the Complement of the Graph

Next, we need to find the complement of the graph \(G\). The complement of a graph is obtained by replacing all of its edges with non-adjacent vertices. In this case, for graph \(G\), since every vertex is connected, \(\bar{G}\) will have no edges and thus also is connected.
03

Part B: Starting the Proof

The proof starts from the given that \(G\) is not connected, which means there exist at least two vertices for which there is no path between them. The task is to prove that \(\bar{G}\) is connected, that is, there exists a path between every pair of vertices.
04

Part B: Providing the Proof

In \(\bar{G}\), every pair of non-adjacent vertices in \(G\) becomes an edge. Therefore, for any pair of vertices in \(\bar{G}\), either they were adjacent in \(G\) or if they were not connected, they are now adjacent in \(\bar{G}\). This means there exists a path between every pair of vertices in \(\bar{G}\), hence proving that if \(G\) is not connected, then \(\bar{G}\) is connected.

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Most popular questions from this chapter

a) For all \(x, y \in \mathbf{Z}^{+}\), prove that \(x^{3} y-x y^{3}\) is even. b) Let \(V=\\{1,2,3, \ldots, 8,9\\} .\) Construct the loop-free undirected graph \(G=(V, E)\) as follows: For \(m, n \in V\), \(m \neq n\), draw the edge \(\\{m, n\\}\) in \(G\) if 5 divides \(m+n\) or \(m-n\) c) Given any three distinct positive integers, prove that there are two of these, say \(x\) and \(y\), where 10 divides \(x^{3} y-x y^{3}\)

For \(n \in \mathbf{Z}^{+}\), how many distinct (though isomorphic) paths of length 2 are there in the \(n\)-dimensional hypercube \(Q_{n}\) ?

a) Determine \(P(G, \lambda)\) for \(G=K_{1,3}\). b) For \(n \in \mathbf{Z}^{+}\), what is the chromatic polynomial for \(K_{1, n}\) ? What is its chromatic number?

If \(G=(V, E)\) is a directed graph or multigraph with no isolated vertices, prove that \(G\) has a directed Euler trail if and only if (i) \(G\) is connected; (ii) \(o d(v)=i d(v)\) for all but two vertices \(x, y\) in \(V\); and (iii) \(o d(x)=i d(x)+1, \operatorname{id}(y)=\operatorname{od}(y)+1\).

a) Let \(k \in \mathbf{Z}^{+}, k \geq 3 .\) If \(G=(V, E)\) is a connected planar graph with \(|V|=v,|E|=e\), and each cycle of length at least \(k\), prove that \(e \leq\left(\frac{k}{k-2}\right)(v-2)\). b) What is the minimal cycle length in \(K_{3,3}\) ? c) Use parts (a) and (b) to conclude that \(K_{3,3}\) is nonplanar. d) Use part (a) to prove that the Petersen graph is nonplanar.
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