When does the complete bipartite graph K n,m have an Euler Trail(Path)?

Question

So I know that an Euler trail must have no more than two odd degree vertices.

So does this mean that either $n$ or $m$ must be odd? Or is it $n = m + 1$?

Answer 1

You're right that it has an Euler trail if and only if the number of odd-degree vertices is at most $2$.

In $K_{m,n}$ there are $m$ vertices of degree $n$ and $n$ vertices of degree $n$.

If $m,n$ are both even then all degrees are even so there is an Euler trail.

If exactly one of them, say $m$, is odd then there are $n$ vertices of odd degree. So when can there be an Euler trail in this case?

If $m,n$ are both odd then there are $m+n$ odd degree vertices. When does this give an Euler trail?