G is a simple undirected graph. Some vertices of G are of odd degree. Add a…

Answer: Euler. The resulting graph is Eulerian because every vertex has even degree.

• By the Handshaking Lemma, the number of vertices of odd degree in any graph is even.

• Joining the new vertex to every odd-degree vertex increases each odd degree by 1, making those degrees even. Vertices that were even remain even.

• The new vertex has degree equal to the number of odd-degree vertices, which is even, so its degree is even as well.

• Therefore every vertex in the resulting graph has even degree, so the graph is Eulerian (each connected component has an Euler circuit).

Why the other answers are incorrect:

• Not necessarily regular: degrees become even but need not be equal across vertices.

• Not necessarily complete: no new edges are added between existing vertex pairs, so completeness is not guaranteed.

• Not necessarily Hamiltonian: even degrees do not ensure a Hamiltonian cycle.