Approximation for metric TSP: Worst case using nearest neighbor heuristic?
No, the NN heuristic does not have constant factor for metric TSP. 

Rosenkrantz, Stearns, and Lewis proved in An Analysis of Several Heuristics for the Traveling Salesman Problem. SIAM J. Comput. 6(3): 563-581 (1977) that the worst case ratio of the tour obtained by the nearest neighbor method is bounded above by a logarithmic function of the number of nodes.