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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



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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.


a

d

e

c

b

G

Illustration:

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:


a

b

d

c

G

e

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