Find the degree of remaining verticies

Question

A simple graph G has 7 vertices and 9 edges. The degrees of some of its vertices are 2, 2, 4, 2. Furthermore, G is known to have an Euler circuit. Find the degrees of the remaining vertices, and draw a picture of G

How can i do this? I'm very tired so clear instruction would be great!

What I have done: To find degrees of the remaining verticies = $2*9$ =18 therefore remaining degrees = $18-10 = 8$

Since G is a Euler circuit the degrees must be even. Therefore 3 remaining verticies must have a degree of 2, 2 and 4.

Answer 1

Hints:

• By the Handshaking Lemma, the sum of the degrees of the remaining vertices is $8$.
• Since $G$ has an Euler circuit, all vertices must have a strictly positive even degree.