I'm trying to prove that a graph with the below properties has an eulerian path. 1. Graph has no self loops or parallel edges. 2. Every two different vertices u and v have an odd number of common neighbor vertices. I'm thinking about this problem for a whole day and can't manage to prove this. I think that I can prove it if I could prove that every graph with these properties must have odd number of vertices but I can't prove it as well.