Question: Design an efficient algorithm that given any node VI in any network N with n nodes produces one of the following three outputs:
(i) a labeling of the nodes as V1, V2,......., Vn such that (19.3) holds;
(ii) a coloring of the nodes as in problem, showing that N is not connected;
(iii) a cycle in N.
Problem: Prove that a network is not connected if and only if its nodes can be colored green and red in such a way that each color occurs at least once but the two endpoints of every arc have the same color.