A graph with Hamilton paths but no Euler paths

hbghlyj

A graph with Hamilton paths but no Euler paths:

The graph has a Hamilton path (many different ones, actually), as shown here:

A graph with Euler paths but no Hamilton paths:

hbghlyj

math.stackexchange.com/questions/3004736/when … e-an-euler-trailpath

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. Only when $n=2$ can there be an Euler trail.

If $m,n$ are both odd then there are $m+n$ odd degree vertices. Only when $n=m=1$ does this give an Euler trail.