Cube versus Torus Models and the Euclidean Minimum Spanning Tree Constant

We show that the length of the minimum spanning tree through points drawn uniformly from the d-dimensional torus is almost surely asymptotically equivalent to the length of the minimum spanning tree through points drawn uniformly from the d-cube. This result implies that the analytical expression recently obtained by Avram and Bertsimas for the MST constant in the d- torus model is in fact valid for the traditional d-cube model. We also show that the number of vertices of degree k for the MST in both models are asymptotically equivalent with probability one. Finally we show how our results can be extended to other combinatorial problems such as the traveling salesman problem