Are there any $K_n$ that have Euler trails but not Euler cycles?

Question

I know that in $K_n$, a complete graph. ALL the vertices must be of even degree for a euler cycle to exist. So n should be odd in this case if we want euler cycles. And if n is even, then $K_n$ has ALL vertices, not just two be of odd degree, so no Euler Trail and Euler Cycle exists. So naturally this causes me to believe that the answer is, no there is no $K_n$ that works here. But what about $K_2$? which is just an edge. Would that work here?

Answer 1

You're correct that a graph has an Eulerian cycle if and only if all its vertices have even degree, and has an Eulerian path if and only if exactly $0$ or exactly $2$ of its vertices have an odd degree. If $n$ is odd, then the first condition is satisfied, and if $n$ is even then all $n$ of the vertices have odd degree. So, this fails both conditions unless $n=2$, so what you've written does not preclude the fact that $K_2$ has an Eulerian path. (In fact, it does, as you can easily see from the definition of Eulerian path.)