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Chapter 11: Problem 2

As the chair for church committees, Mrs. Blasi is faced with scheduling the meeting times for 15 committees. Each committee meets for one hour each week. Two committees having a common member must be scheduled at different times. Model this problem as a graph-coloring problem, and tell how to determine the least number of meeting times Mrs. Blasi has to consider for scheduling the 15 committee meetings.

Short Answer

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To determine the least number of meeting times Mrs. Blasi has to consider for scheduling the 15 committee meetings, the problem needs to be modelled as a graph-coloring problem. The least number of meeting times is then essentially the chromatic number of the respective graph, which is the minimum number of colors needed to color the graph such that no two adjacent vertices share the same color. This number gives the least number of distinct meeting times needed to make sure that no committee members have scheduling conflicts.

Step by step solution

01

Problem Representation

Represent the committee meetings as a graph. Each committee becomes a vertex and if there is a member who belongs to two different committees, draw an edge between those two vertices. Each vertex of the graph now represents a committee, and each edge represents the conflict of scheduling, i.e., if two committee meetings are connected via an edge, they must be scheduled at different times.
02

Graph Coloring

The coloring of the graph implies the scheduling of the committee meetings. Using a graph coloring algorithm, assign colors to the vertices. Remember that two adjacent vertices (vertices that share an edge) should not share the same color. This process continues until you've assigned a color (meeting time) to every committee (vertex). Assign the smallest color that has not been used on any vertices adjacent to it.
03

Determine the Least Number of Meeting Times

The least number of meeting times Mrs. Blasi has to consider for scheduling the committee meetings is the chromatic number of the graph 鈥� which is the fewest colors needed to color a graph. This number gives the least number of distinct meeting times committees need to meet such that no two committees with a common member have the same meeting time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chromatic Number
Graph coloring is an important concept in scheduling problems like the one faced by Mrs. Blasi. When we discuss the chromatic number, we're talking about the smallest number of colors needed to color the vertices of a graph such that no two adjacent vertices share the same color. This scenario is directly applicable to scheduling meeting times where each committee is represented by a vertex and any shared members result in an edge between vertices.
  • The chromatic number helps determine the number of different time slots required for all meetings.
  • A graph鈥檚 chromatic number is the key to resolving scheduling conflicts, minimizing overlap.
  • To compute the chromatic number, a systematic graph coloring algorithm is used.
Calculating the chromatic number provides a practical solution to Mrs. Blasi's scheduling dilemma, ensuring the least number of time slots are used without scheduling conflicts. By applying this number, she can efficiently plan the meeting times so that no one is double-booked.
Vertex Coloring
Vertex coloring involves assigning colors to vertices of a graph in a way that no two adjacent vertices share the same color. In the case of committees, each vertex is a committee, and edges represent shared members. To implement vertex coloring:
  • Assign an initial color to a vertex.
  • Move on to adjacent vertices, assigning different colors as necessary.
  • Continue until every vertex has a color that doesn't match an adjacent vertex.
Using vertex coloring efficiently assigns time slots or schedules for committee meetings. This ensures that committees sharing members do not have overlapping meetings. The challenge lies in finding the minimal set of colors. Doing so means Mrs. Blasi can find the perfect schedule with the fewest time slots possible, optimizing the scheduling of the 15 committees.
Conflict Resolution
Conflict resolution in this context refers to resolving scheduling conflicts among committees with common members. The goal is to ensure that all members can attend their respective committees without any overlap. When conflicts are represented by edges connecting vertices, resolving them means finding a coloring with no two adjacent vertices having the same color. To handle conflict resolution:
  • Use graph coloring techniques to uniquely color each vertex.
  • Ensure that adjacent vertices have different colors or, in other words, different meeting times.
  • Through this, adjacent vertices signify committees with shared members and need resolved meeting schedules.
Resolving conflicts through this method ensures efficient use of time and resources. Mrs. Blasi can then confidently assign meeting slots knowing no individual faces double bookings due to shared committee memberships.

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Most popular questions from this chapter

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.

Give an example of an undirected graph \(G=(V, E)\), where \(\chi(G)=3\) but no subgraph of \(G\) is isomorphic to \(K_{3}\).

Let \(n=2^{k}\) for \(k \in \mathbf{Z}^{+}\). We use the \(n k\)-bit sequences (of 0 's and 1 's) to represent \(1,2,3, \ldots, n\), so that for two consecutive integers \(i, i+1\), the corresponding \(k\)-bit sequences differ in exactly one component. This representation is called a Gray code (comparable to what we saw in Example 3.9). a) For \(k=3\), use a graph model with \(V=\\{000,001\), \(010, \ldots, 111\\}\) to find such a code for \(1,2,3, \ldots, 8\). How is this related to the concept of a Hamilton path? b) Answer part (a) for \(k=4\).

If \(G=(V, E)\) is an undirected graph, a subset \(D\) of \(V\) is called a dominating set if for all \(v \in V\), either \(v \in D\) or \(v\) is adjacent to a vertex in \(D\). If \(D\) is a dominating set and no proper subset of \(D\) has this property, then \(D\) is called minimal. The size of any smallest dominating set in \(G\) is denoted by \(\gamma(G)\) and is called the domination number of \(G\). a) If \(G\) has no isolated vertices, prove that if \(D\) is a minimal dominating set, then \(V-D\) is a dominating set. b) If \(I \subseteq V\) is independent, prove that \(I\) is a dominating set if and only if \(I\) is maximal independent. c) Show that \(\gamma(G) \leq \beta(G)\), and that \(|V| \leq \beta(G) \chi(G)\). [Here \(\beta(G)\) is the independence number of \(G-\) first given in Exercise 25 of Section 11.5.]

a) At the J. \& J. Chemical Company, Jeannette has received three shipments that contain a total of seven different chemicals. Furthermore, the nature of these chemicals is such that for all \(1 \leq i \leq 5\), chemical \(i\) cannot be stored in the same storage compartment as chemical \(i+1\) or chemical \(i+2\). Determine the smallest number of separate storage compartments that Jeannette will need to safely store these seven chemicals. b) Suppose that in addition to the conditions in part (a), the following four pairs of these same seven chemicals also require separate storage compartments: 1 and 4,2 and 5,2 and 6 , and 3 and 6 . What is the smallest number of storage compartments that Jeannette now needs to safely store the seven chemicals?
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