I don't have an answer for this exercise, so I'm asking you.
Exercise: Draw a graph with 6 vertices with the degrees: 2, 3, 3, 4, 4, 4. Name the vertices a, b, c, d, e, f (in the order you want), and then find an Euler path in your graph. My solution :
It looks correct. Here's a drawing where I've added edge directions and increasingly blue colors according to the Eulerian path, and identified the vertex degrees (ignoring edge directions).