To find a Hamilton cycle is a NPC problem, but Euler is not. Considering one can always transform the vertex as edge or vice versa conceptually. Then the vertex can be used to describe the information which originally edge does.
What properties make the Euler graph more easily to be resolved?
For example, in genome assembly problem, one can either consider a Kmer as vertex or edge(de bruijn graph), it's just two perspectives to look at the same thing. I think their should be additional or missing information between two kind of explanation.