COSMO · STUDY + TEACHING DECK

Networks

January 11, 2026

A question worth sharing

Graph Coloring

Assign limited labels so adjacent objects never conflict.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use proper coloring, chromatic number, clique to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

How many colors does an odd cycle require?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

proper coloring

adjacent vertices receive different colors

chromatic number

fewest colors needed

clique

set of pairwise adjacent vertices

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

G\text{ is bipartite}\quad\Longleftrightarrow\quad G\text{ has no odd cycle}

The idea

Assign limited labels so adjacent objects never conflict.

What to remember

A coloring turns pairwise conflicts into a global resource plan.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Color a six-cycle with the fewest colors.

Predict each next step

Alternate colors A and B around the cycle.
The even length returns consistently.
Every edge joins different colors.
Two colors are necessary and sufficient.

Which step needs a reason, not just a calculation?

See the structure

The structure behind graph coloring

G\text{ is bipartite}\quad\Longleftrightarrow\quad G\text{ has no odd cycle}

Two alternating colors close on an even cycle; an odd cycle forces a conflict.

Read the representation

Assign limited labels so adjacent objects never conflict.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate graph coloring

Your shared task

Teams model a scheduling conflict graph and compare greedy colorings from different vertex orders.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A coloring turns pairwise conflicts into a global resource plan.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Explain why a graph containing $K_{4}$ needs at least four colors.

Predict each next step

Every pair of $K_{4}$ vertices is adjacent.
No two vertices may share a color.
Four distinct vertices need four distinct colors.
Thus the chromatic number is at least four.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A graph is bipartite exactly when it contains no odd cycle.

Build the argument

A two-coloring cannot close consistently around an odd cycle.
If no odd cycle exists, breadth-first levels by parity give two parts.
Edges never join vertices with the same parity level.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Find the chromatic number of the wheel formed from a five-cycle plus a center adjacent to every cycle vertex.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Coloring models exam schedules, frequency assignment, registers, and map constraints.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

How many colors does an odd cycle require?

Question 2

Explain why this claim is true: A graph is bipartite exactly when it contains no odd cycle.

Question 3

Find the chromatic number of the wheel formed from a five-cycle plus a center adjacent to every cycle vertex.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A coloring turns pairwise conflicts into a global resource plan.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.