Euler path and Euler circuit problem

Sep 2009
For which n does \(\displaystyle K_n\) have an Euler path but not an Euler circuit?
Apr 2010
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?