Tools

Graph Colouring

Graph Coloring Experiment Procedure

Objective

To understand the concept of graph coloring by interactively coloring vertices of a graph such that no two adjacent (connected) vertices share the same color, using the minimum number of colors possible.

Prerequisites

  • Basic understanding of graphs (vertices and edges)
  • Familiarity with the concept of adjacency in graphs

Getting Started with the Simulation

Step 1: Understanding the Interface

  1. Graph Display Area: The main canvas shows a randomly generated graph with vertices (circles) and edges (lines connecting vertices)
  2. Color Palette: Located below the challenge section, displays available colors for vertex coloring
  3. Control Buttons:
    • "Check Coloring" - Validates your current coloring attempt
    • "Show Solution" - Displays an automatic optimal coloring
    • "New Graph" - Generates a different random graph
    • "Clear Colors" - Removes all vertex colors to start fresh

Step 2: Customizing Graph Parameters (Optional)

  1. Click the floating controls button (⚙️ icon) in the bottom-right corner
  2. Adjust the following parameters using sliders:
    • Number of Vertices: Choose between 4-10 vertices
    • Number of Colors: Select 2-6 available colors
  3. Observe how the Graph Info section updates with:
    • Total number of vertices and edges
    • Chromatic number (minimum colors needed)

Interactive Coloring Procedure

Step 3: Color Selection

  1. Select a Color: Click on any color from the color palette
    • The selected color will be highlighted with a darker border
    • Available colors are displayed as colored circles
    • A clear option (×) allows you to remove colors from vertices

Step 4: Vertex Coloring Process

  1. Choose a Starting Vertex: Click on any vertex in the graph

    • The vertex will be colored with your selected color
    • Vertex labels (A, B, C, etc.) help identify each vertex

  2. Apply Strategic Coloring:

    • Rule: No two adjacent vertices can have the same color
    • Strategy: Start with vertices that have the most connections (highest degree)
    • Tip: Try to use as few colors as possible for an optimal solution

  3. Continue Coloring:

    • Select different colors from the palette as needed
    • Color remaining vertices while respecting the adjacency constraint
    • Use the visual feedback to identify connections between vertices

Step 5: Solution Validation

  1. Check Your Work: Click the "Check Coloring" button

    • Success Message: "Excellent! Valid coloring using X colors."
    • Error Messages:
      • "Please color all vertices!" (if some vertices remain uncolored)
      • "Invalid! Vertices X and Y are adjacent and have the same color." (if constraint violated)

  2. Feedback Interpretation:

    • Green feedback indicates a valid coloring
    • Red feedback indicates errors that need correction
    • The system checks both completeness and correctness

Step 6: Learning from Automatic Solutions

  1. View Optimal Solution: Click "Show Solution" to see an algorithmic coloring
  2. Compare Results: Observe how the automatic solution differs from your attempt
  3. Analyze Efficiency: Note the number of colors used in the optimal solution

Advanced Exploration

Step 7: Experimenting with Different Graphs

  1. Generate New Challenges: Use "New Graph" to practice with different graph structures
  2. Vary Complexity: Adjust vertex count to experience graphs of different sizes
  3. Color Constraints: Experiment with different numbers of available colors

Step 8: Understanding Graph Properties

  1. Observe Patterns:

    • Complete Graphs: Require as many colors as vertices
    • Bipartite Graphs: Can be colored with only 2 colors
    • Trees: Always require exactly 2 colors

  2. Chromatic Number Analysis:

    • Compare your color usage with the theoretical minimum
    • Understand how graph structure affects coloring complexity

Learning Outcomes

Upon completion of this experiment, you will be able to:

  1. Apply the graph coloring constraint practically
  2. Develop strategies for efficient vertex coloring
  3. Understand the relationship between graph structure and chromatic number
  4. Recognize real-world applications of graph coloring problems

Troubleshooting Tips

  • Mobile Users: The interface is optimized for touch interaction
  • Color Selection: Ensure a color is selected before clicking vertices
  • Visual Clarity: Use the floating info panel (ℹ️) for additional guidance
  • Reset Option: Use "Clear Colors" to start over without generating a new graph