Activity: Exploring Graph Coloring through Play and Challenges

Objective: To deepen understanding of graph coloring by interactive play and progressively challenging problems.

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Introduction

Begin by having the students play with the online graph coloring game at:

Graph Coloring Game

• Instructions:
  • Explore the game by attempting to color various graphs with the smallest number of colors.
  • Try to solve all the challenges provided in the game.
  • Pay attention to how the number of colors relates to the structure of the graph.

This interactive experience will provide a hands-on understanding of graph coloring concepts and prepare them for the subsequent challenges.

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After the Game: Deeper Exploration

Step 1: Recall Graph Coloring Basics

• Definition: In a graph, a proper vertex coloring assigns colors to each vertex such that no two adjacent (connected) vertices share the same color.
• Chromatic Number (χ): The smallest number of colors needed to properly color a graph.

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Step 2: Investigate Small Graphs

Ask the students to observe chromatic numbers in small graphs:

1. Graph with 1 Vertex (V = 1):

   • Edges (E): 0
   • Chromatic Number (χ): 1

2. Graph with 2 Vertices (V = 2):

   • If connected (edge between them):
     • Chromatic Number (χ): 2

3. Graph with 3 Vertices (V = 3):

   • Triangle (each vertex connected to the other two):
     • Chromatic Number (χ): 3

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Step 3: Identify the Pattern

• Observation: In a complete graph, the chromatic number equals the number of vertices.
• Complete Graph (Kₙ): A graph where every pair of distinct vertices is connected by a unique edge.

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Step 4: Generalize the Concept

• Conclusion: A graph where the chromatic number equals the number of vertices is a complete graph.
• Reasoning: Each vertex is adjacent to every other vertex, so each must have a unique color.

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Step 5: Draw and Analyze

Have the students:

1. Draw Complete Graphs for V = 4, 5, 6:

   • Label vertices and connect every pair of vertices.

2. Attempt to Color with Fewer Colors:

   • Try to color the graph with fewer colors than vertices and observe conflicts.

3. Discuss Why It's Impossible:

   • Emphasize adjacency requiring different colors.

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Step 6: Extend the Exploration

• Non-Complete Graphs: Find graphs where the chromatic number is less than the number of vertices.

  • Example: A bipartite graph or a cycle graph with an even number of vertices.

• Real-World Applications: Discuss applications like scheduling, map coloring, and frequency assignment.

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10 Challenge Questions on Graph Coloring

After playing the game and exploring basic concepts, challenge the students with the following problems:

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Challenge 1: Basic Coloring

Question: Draw a simple graph with 4 vertices that can be properly colored using only 2 colors. What kind of graph is this?

Hint: Think of a square or a cycle graph with 4 vertices.

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Challenge 2: Coloring a Cycle Graph

Question: Consider a cycle graph with 5 vertices. What is the minimum number of colors needed to color this graph properly?

Hint: Observe the parity of the cycle (odd number of vertices).

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Challenge 3: Understanding Bipartite Graphs

Question: Explain why any bipartite graph can be colored using only 2 colors, regardless of the number of vertices.

Hint: Bipartite graphs have vertices divided into two sets with edges only between sets.

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Challenge 4: Complete Graph Coloring

Question: How many colors are needed to properly color a complete graph with 4 vertices (( K_4 ))? Draw the graph and show your coloring.

Answer: 4 colors, because each vertex is connected to every other vertex.

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Challenge 5: Chromatic Number Exploration

Question: Find a graph with 6 vertices that requires exactly 3 colors to be properly colored. Describe or draw your graph.

Hint: Think of two interconnected triangles.

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Challenge 6: Coloring Planar Graphs

Question: According to the Four Color Theorem, any planar graph (a graph that can be drawn on a plane without edges crossing) can be colored using at most 4 colors. Can you draw a planar graph with 5 vertices that requires exactly 4 colors?

Hint: Consider the complete graph ( K_5 ) minus one edge, making it planar.

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Challenge 7: Graph with Chromatic Number Less Than Clique Number

Question: Can you find a graph where the chromatic number is less than the size of its largest complete subgraph (clique)? Draw the graph and explain.

Hint: Construct a graph with a large clique embedded in a structure that reduces the overall chromatic number.

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Challenge 8: Trees and Coloring

Question: Prove that any tree can be properly colored using only 2 colors, regardless of its size.

Hint: Use induction or consider the properties of acyclic graphs.

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Challenge 9: Graph Coloring and Map Coloring

Question: Is it possible to draw a map such that the corresponding graph requires 5 colors to be properly colored? Explain your reasoning.

Answer: No, due to the Four Color Theorem, any planar graph (and thus any map) can be colored with at most 4 colors.

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Challenge 10: Chromatic Number and Graph Properties

Question: Create a graph with 7 vertices that has a chromatic number of 4. Explain the properties that necessitate the use of 4 colors.

Hint: Incorporate an odd cycle or a complete subgraph of 4 vertices (( K_4 )).

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Additional Tips for Students

• Engage with the Game: Reflect on strategies used in the online game and how they apply to these challenges.
• Draw and Experiment: Visualizing the graphs will aid in understanding.
• Collaborate: Discuss your approaches with peers.
• Explore Variations: Modify graphs to see how changes affect the chromatic number.

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By integrating interactive play with these challenges, students will build a solid foundation in graph coloring concepts and develop problem-solving skills in graph theory.