we wil now show that a connected multigraph has an euler path (and not an euler circuit)
if and only if it has exactly two vertices of odd degree. first, suppose that a connected multigraph does have an euler path from a to d, but not an euier circuit. the first edge of the path contributes one to the degree of a. A contribution of two to the degree of a is made every time contributes passes through a. the last edge in the path contributes one to the degree of b. every time the path goes through b there is contribution of two its degree. consequently, both a and b have odd degree. every other vertex has even degree, the path contributes two to the degree of a vertex whenever it passes through it.