Study Notes on Graph Theory Applications in Frequency Allocation and Exam Scheduling

Coloring Applications

  • Overview of Frequency Channel Allocation

    • Signals of six radio frequency stations are monitored to ensure no interference.

    • The Federal Communications Commission (FCC) oversees the allocation of appropriate frequencies to each station.

    • Objective: Determine the minimum number of different frequency channels for six stations.

    • Condition: Two stations can share the same channel if they are more than 150 miles apart.

Distance Table

  • Station Distances: Below is the distance table between the stations:

    • SA (Station A)

    • SB: 25 miles

    • SC: 202 miles

    • SD: 77 miles

    • SE: 375 miles

    • SF: 106 miles

    • SB (Station B)

    • SA: 25 miles

    • SC: 175 miles

    • SD: 51 miles

    • SE: 148 miles

    • SF: 222 miles

    • SC (Station C)

    • SA: 202 miles

    • SB: 175 miles

    • SD: 111 miles

    • SE: 365 miles

    • SF: 411 miles

    • SD (Station D)

    • SA: 77 miles

    • SB: 51 miles

    • SC: 111 miles

    • SE: 78 miles

    • SF: 297 miles

    • SE (Station E)

    • SA: 375 miles

    • SB: 148 miles

    • SC: 365 miles

    • SD: 78 miles

    • SF: 227 miles

    • SF (Station F)

    • SA: 106 miles

    • SB: 222 miles

    • SC: 411 miles

    • SD: 297 miles

    • SE: 227 miles

Graph Coloring

  • Vertex Coloring Technique:

    • The goal is to ensure that no two stations that are within 150 miles of each other use the same frequency, while those over 150 miles apart can share channels.

    • The following groupings show which stations can use the same channel:

    • SA can share channels with: SB, SD, SF

    • SB can share channels with: SA, SD, SE

    • SC can share channels with: SD

    • SD can share channels with: SA, SB, SE

    • SE can share channels with: SB, SD

    • SF can share channels with: SA, SB, SC, SD, SE

  • Implication of Graph Coloring:

    • The method provides the chromatic number of the graph representing our stations, indicating the minimal frequency channels required.

    • Two valid colorings were identified:

    • Coloring 1: {SA, SE}, {SB, SC}, {SD, SF}

    • Coloring 2: {SA, SE, SC}, {SB}, {SF, SD}

Conclusion on Frequency Channels

  • Final Assessment:

    • Based on the vertex coloring results from Coloring 1 and Coloring 2, at least three different frequency channels are required for the six stations to ensure their signals do not interfere.

Exam Scheduling Problem

  • Context of Exam Scheduling:

    • Four courses of study: Algebra, Physics, Statistics, and Calculus.

    • Common students exist between the following pairs:

    • Algebra and Statistics

    • Algebra and Calculus

    • Statistics and Physics

  • Core Problem Statement:

    • If the Algebra and Statistics exams are scheduled on the same day, students enrolled in both will face a conflict and may miss one exam.

    • The challenge is to find a way to schedule exams over the minimum number of days while ensuring that courses with common students do not conflict in their scheduling.

Solution via Graph Coloring for Exam Scheduling

  • Graph Construction:

    • A graph is created where each course serves as a vertex.

    • An edge connects two vertices (courses) if there are common students between them.

  • Graph Coloring Approach:

    • The objective is to assign colors (exam days) to the vertices so that no two adjacent vertices (courses with common students) share the same color. This ensures that no conflicting exams are on the same day as shown in an illustrative example that wasn't provided in the transcript.