Step by step
The question
Question: find an Euler path in the graph AB, BC, CD, AD.
1 Draw the graph
Each letter is a vertex: a dot.
Each pair is an edge: a line between two vertices.
An Euler path uses every edge exactly once.
An Euler circuit also ends where it started.
2 Count the degrees
The degree of a vertex is how many edges touch it.
A has 2.
B has 2.
And so on.
3 Add the degrees
Add the degrees: 8.
That is 2 times the 4 edges, because each edge has two ends.
It always works: the handshake rule.
4 Find the odd vertices
The rule: each time we pass through a vertex we use two of its edges, one in and one out.
So only the start and the end can be odd.
Careful!
With no odd vertices we can start anywhere, and we come back to the start.
5 Walk the route
Start at A.
Go to B: edge 1.
Last, D to A: edge 4.
The circuit: A, B, C, D, A.
6 Check
Check: every edge has exactly one number, 1 to 4, and there are 4 edges.
Right!
Let's look back.
First: draw the graph.
Then: count the degrees.
Then: add the degrees.
Then: find the odd vertices.
Then: walk the route.
Last: check.