The following table shows the list of the math exams with student IDs
registered for each course. We want to schedule the exams so that two
exams with common students don’t fall on the same day. What is the
minimum number of days we need to schedule these exams?
Math 101: 2001 2003 2004 2008
Math 201: 2002 2003 2008
Math 232: 2002 2005 2007 2006
Math 241: 2003 2004 2006
Math 344: 2001 2005


We can approach this problem as a **graph coloring problem**. The goal is to assign colors (days) to the exams (vertices) such that no two exams with common students (edges between vertices) are scheduled on the same day.

### Step 1: Build a Graph

- Each exam is represented as a vertex.
- Draw an edge between two vertices (exams) if they have a student in common.

Here’s how the graph would look based on the student overlap:

- **Math 101 and Math 201**: Student 2003 and 2008 are common.
- **Math 101 and Math 241**: Student 2003 and 2004 are common.
- **Math 101 and Math 344**: Student 2001 is common.
- **Math 201 and Math 241**: Student 2003 is common.
- **Math 201 and Math 232**: Student 2002 is common.
- **Math 232 and Math 241**: Student 2006 is common.
- **Math 232 and Math 344**: Student 2005 is common.

### Step 2: Draw the Adjacency Graph
The graph will have 5 vertices corresponding to the 5 exams, and edges based on the common students between exams.

### Step 3: Determine the Minimum Number of Days
This corresponds to finding the **chromatic number** of the graph (the minimum number of colors needed to color the graph).

- If we color the graph, we must ensure that connected vertices (exams with common students) do not have the same color (day).

Let me compute and explain the chromatic number of this graph to find the minimum number of days required.

I'll draw the graph and determine the minimum number of colors (days) needed.import networkx as nx

# Create a graph
G = nx.Graph()

# Add nodes representing the exams
exams = ["Math 101", "Math 201", "Math 232", "Math 241", "Math 344"]
G.add_nodes_from(exams)

# Add edges based on student overlap between exams
edges = [
    ("Math 101", "Math 201"),  # Students 2003 and 2008 in common
    ("Math 101", "Math 241"),  # Students 2003 and 2004 in common
    ("Math 101", "Math 344"),  # Student 2001 in common
    ("Math 201", "Math 241"),  # Student 2003 in common
    ("Math 201", "Math 232"),  # Student 2002 in common
    ("Math 232", "Math 241"),  # Student 2006 in common
    ("Math 232", "Math 344")   # Student 2005 in common
]
G.add_edges_from(edges)

# Find the chromatic number (minimum number of colors needed)
chromatic_number = nx.coloring.greedy_color(G, strategy="largest_first")
num_colors_needed = max(chromatic_number.values()) + 1

num_colors_needed, chromatic_numberThe minimum number of days required to schedule the exams is **3**. 

Here is how the exams can be scheduled across the 3 days (represented by colors):

- **Day 1**: Math 101, Math 232
- **Day 2**: Math 201, Math 344
- **Day 3**: Math 241

No two exams with common students are scheduled on the same day. 

Would you like further details on how the graph was colored or any clarifications?

Here are 5 questions for deeper thought:
1. What if we added another exam with a new set of students? How would it affect the scheduling?
2. How would the solution change if certain exams must be scheduled on specific days?
3. Can we always solve this problem using graph coloring? What are the exceptions?
4. What is the relationship between the chromatic number and the complexity of scheduling problems?
5. Can this technique be applied to scheduling beyond exams, such as work shifts or sports events?

**Tip**: Graph coloring is a versatile technique for solving problems that involve conflicts, such as scheduling tasks or map coloring.

Learn how to minimize conflicts in scheduling exams using graph coloring. Explore the application of graph theory concepts like graph coloring and chromatic number in solving real-world scheduling problems.

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