Theorem 1: A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has even degree

Theorem 1: A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has even degree.

• Here a,b,c,d,e are vertices.
• Each of the vertices have even degree.
• Hence the graph has an Euler circuit namely a,d,c,e,d,b,a.

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.

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.

(Note: Those two vertices are the end point of the path formed.)

Illustration:

• Here, a, b, c, d, e are vertices.
• The vertices d, c and e have even degree and the vertices a and b have odd degree.
• Since the graph G has exactly two vertices with odd degree, it has an Euler path, namely, a, d, c, e, b, c, a, b ( a and b are end points ) but not an Euler circuit.