COSMO · STUDY + TEACHING DECK

Networks

October 3, 2025

A question worth sharing

Graph Theory

Routes, degree, and the moment a drawing becomes an argument.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use vertex, edge, degree 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

A connected network has edges AB, BC, CD, DE, EF, FA, CF, BF, and CE. Can you trace every edge exactly once? Where must you start?

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

Count how many edges touch each vertex.
Circle vertices with odd degree.
Compare the odd vertices with possible start and end points.

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.

vertex

a location or object

edge

a connection

degree

the number of edge-ends at a vertex

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

The core idea

Keep the idea. Explain the conditions.

\sum_{v\in V}\deg(v)=2|E|

The idea

Routes, degree, and the moment a drawing becomes an argument.

What to remember

Count edge-ends locally to reason about an entire network.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Why do internal vertices in a one-stroke route usually have even degree?

Predict each next step

Enter along one unused edge.
Leave along a different unused edge.
Those edges pair as enter and exit.
Only the start and end may remain unpaired.

Which step needs a reason, not just a calculation?

See the structure

Degree predicts whether a route can exist

B and E are the only vertices with odd degree.

Read the representation

A connected graph has an Euler circuit with zero odd vertices and an Euler path with exactly two odd vertices.

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.

Euler path

Uses every edge exactly once.
May revisit vertices.
Controlled by odd-degree count.

Hamilton path

Visits every vertex exactly once.
May skip edges.
Not decided by the same simple degree rule.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Visits every vertex exactly once.
May skip edges.
Not decided by the same simple degree rule.

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

Design and trade route puzzles

Your shared task

Each group draws a connected graph with 0, 2, or 4 odd vertices, then another group predicts traceability.

Explore

5 min - draw and count

Compare

5 min - swap and predict

Defend

4 min - verify by tracing

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

The handshaking sum equals twice the number of edges.
Therefore the number of odd-degree vertices must be even.
A local count at vertices can decide a global route question.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Can a graph have exactly one odd-degree vertex?

Predict each next step

Sum all vertex degrees.
Every edge contributes 2 to the sum.
The total degree is even.
An odd number of odd terms cannot make an even total.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Every tree with at least two vertices has at least two leaves.

Build the argument

Take a longest path in the tree.
An endpoint cannot have another unused neighbor.
Otherwise the path could be extended, so both endpoints have degree 1.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

A museum has rooms joined by doors. What graph would model a route that uses every door once? What would vertices represent?

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

Transit routes, circuit inspection, communication networks, dependencies, and scheduling become clearer when objects and connections are separated.

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

Why can a graph not have exactly three odd vertices?

Question 2

What is the difference between an Euler path and an Euler circuit?

Question 3

How does degree turn a drawing into an argument?

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

Count edge-ends locally to reason about an entire network.

Classify the worksheet networks before tracing them; prediction should come before trial.

A good question belongs to everyone in the room.