# Euler path and Euler circuit problem

#### melody

For which n does $$\displaystyle K_n$$ have an Euler path but not an Euler circuit?

#### Xian

There can be but one...

Excluding the trivial edgeless case of $$\displaystyle K_1$$ we can proceed as follows:

If n is odd
Then $$\displaystyle K_n$$ has all even degrees, and so by a theorem of Euler's we have that there exist an Eulerian circuit, which by technicality admits an Eulerian path (a circuit is a kind of path).

If n is even
Well clearly $$\displaystyle K_2$$ contains an Eulerian path but not an Euler circuit. However for all even n>2 we know that they have more than 2 vertices of odd degree (for $$\displaystyle K_n$$ we have all vertices with degree n-1). For there to be an Eulerian path, we can have at MAX two vertices of odd degree, and for even n>2, this condition fails. And so $$\displaystyle K_2$$ is the only graph the has an Eulerian path but no circuit.

For more info on Eulerian paths and circuits check out How can we tell if a graph has an Euler path or circuit?