Necessary and sufficient conditions for Euler circuits
Obviously a non-connected graph cannot have an Euler path unless it
Theorem 1. A connected multigraph has an Euler circuit if and only if
each of its vertices has even degree.
Assume the graph has an Euler circuit.
Observe that every time the circuit passes through a vertex, it
contributes 2 to the vertex’s degree, since the circuit enters via an edge
incident with this vertex and leaves via another such edge.