A four-vertex cycle has the property that its Euler tour is unique, up to the choice of which vertex you start at and which way you go around the tour. That is, if the four vertices are abcd, you could tour them starting at a as abcda or adcba, starting at b as bcdab or badcb, etc., but those are really all the same tour, because they all arrange the vertices into the same cyclic sequence. Find a four-vertex multigraph that is not a cycle but that has an Euler tour with the same property, that (up to choice of start or reversal of the ordering) there is only one cycle of vertices that forms an Euler tour.