# Euler path and Euler circuit problem

• May 19th 2010, 06:23 PM
melody
Euler path and Euler circuit problem
For which n does $K_n$ have an Euler path but not an Euler circuit?
• May 19th 2010, 09:18 PM
Xian
There can be but one...
Excluding the trivial edgeless case of $K_1$ we can proceed as follows:

If n is odd
Then $K_n$ 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 $K_2$ 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 $K_n$ 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 $K_2$ 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?