An Eulerian cycle, also called an Eulerian circuit, Euler circuit, Eulerian tour, or Euler tour
         
                            
An Euler circuit in a graph in a simple circuit containing every edge of Graph.
An Euler Path in Graph is a simple path containing every edge of graph.
Theorem 1 
A connected multigraph with at least two vertices has an Euler if and only if each of its  vertices has even degree.
Theorem 2
A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.
                           (DISCRETE MATHEMATICS AND ITS APPLICATIONS by Kenneth H.Rosen)