/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Let \(G=(V, E)\) be a connected ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 3

Let \(G=(V, E)\) be a connected undirected graph. a) What is the largest possible value for \(|V|\) if \(|E|=19\) and \(\operatorname{deg}(v) \geq 4\) for all \(v \in V\) ? b) Draw a graph to demonstrate each possible case in part (a).

Short Answer

Expert verified
Given that the degree of every vertex in the graph is at least 4 and there are 19 edges, the largest possible number of vertices would be 9. To illustrate this, one could draw a nonagonal (9-sided) graph where each vertex is connected to its four nearest neighbours (two on each side), creating 19 edges.

Step by step solution

01

Determine the lower bound for vertex count

Given that the degree \(\operatorname{deg}(v)\) must be 鈮� 4, this would imply that the minimum number of vertices \(|V|\) would be 5. This is because, with fewer than five vertices, it is impossible to have each vertex connect to four others.
02

Compute vertex count for given edge count

Assuming a fully connected graph where each vertex connects to all other vertices, the total number of edges \(|E|\) can be calculated using the formula \(|E|= {|V|\choose2}\). However, in this problem, considering the degree \(\operatorname{deg}(v) \geq 4\), the relation \(2|E| = \sum_{v \in V} \operatorname{deg}(v)\) is preferred, implying that every edge is being counted twice, once for each of its end vertices. Since \(\operatorname{deg}(v) \geq 4\), total number of degrees will be \(4|V|\). So, \(2|E| = 4|V|\) or \(|V| = \frac{|E|}{2}\) which calculates to 9.5. Taking an integer value, the highest possible count would thus be 9.
03

Draw the graph

As it has been established that there can be 9 vertices, aim for creating a graph in which each vertex connects to at least four others. This can be accomplished by placing the vertices on a circle and connecting each to its four neighbors (two on each side). It is revealed that such a graph will indeed have 19 edges as specified in the problem.

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Most popular questions from this chapter

Seven towns \(a, b, c, d, e, f\), and \(g\) are connected by a system of highways as follows: (1) I-22 goes from \(a\) to \(c\), passing through \(b ;\) (2) I-33 goes from \(c\) to \(d\) and then passes through \(b\) as it continues to \(f ;(3)\) I-44 goes from \(d\) through \(e\) to \(a ;\) (4) \(\mathrm{I}-55\) goes from \(f\) to \(b\), passing through \(g\); and (5) I-66 goes from \(g\). to \(d\). a) Using vertices for towns and directed edges for segments of highways between towns, draw a directed graph that models this situation. b) List the paths from \(g\) to \(a\). c) What is the smallest number of highway segments that would have to be closed down in order for travel from \(b\) to \(d\) to be disrupted? d) Is it possible to leave town \(c\) and return there, visiting each of the other towns only once? e) What is the answer to part (d) if we are not required to return to \(c ?\) f) Is it possible to start at some town and drive over each of these highways exactly once? (You are allowed to visit a town more than once, and you need not return to the town from which you started.)

Give an example of a connected graph \(G\) where removing any edge of \(G\) results in a disconnected graph.

If \(G\) is an undirected graph with \(n\) vertices and \(e\) edges, let \(\delta=\min _{v \in V}\\{\operatorname{deg}(v)\\}\) and let \(\Delta=\max _{v \in V}\\{\operatorname{deg}(v)\\}\). Prove that \(\delta \leq 2(e / n) \leq \Delta\)

A pet-shop owner receives a shipment of tropical fish. Among the different species in the shipment are certain pairs where one species feeds on the other. These pairs must consequently be kept in different aquaria. Model this problem as a graph-coloring problem, and tell how to determine the smallest number of aquaria needed to preserve all the fish in the shipment.

Let \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\), and let the vertex set \(V^{\prime}\) for the complete graph \(K_{n-1}\) be \(\left\\{v_{1}, v_{2}, v_{3}, \ldots, v_{n-1}\right\\}\). Now construct the loop-free undirected graph \(G_{n}=(V, E)\) from \(K_{n-1}\) as follows: \(V=V^{\prime} \cup\\{v\\}\), and \(E\) consists of all the edges in \(K_{n-1}\) except for the edge \(\left\\{v_{1}, v_{2}\right\\}\), which is replaced by the pair of edges \(\left\\{v_{1}, v\right\\}\) and \(\left\\{v, v_{2}\right\\}\). a) Determine \(\operatorname{deg}(x)+\operatorname{deg}(y)\) for all nonadjacent vertices \(x\) and \(y\) in \(V\). b) Does \(G_{n}\) have a Hamilton cycle? c) How large is the edge set \(E\) ? d) Do the results in parts (b) and (c) contradict Corollary \(11.6 ?\)
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