Theorem: Let G be a connected graph, then the following are equivalent
(i) each vertex of G has even degree
(ii)G has some cycles which between them use each edge of G once and only once
(iii) G is Eulerian
i don't quite get how to show (i)$\iff$ (ii). Also, it seems that (ii)$\implies$(i) is not quite right in general
see,there is cycle which use once and only once each of the edges of G but there are vertexes which are odd degree.