The shortest path and the shortest road through n points

Consider a set of n points lying in a square of side 1. Verblunsky has shown that, if n is sufficiently large, there is some path through all n points whose length does not exceed (2·8 n ) 1/2 +2. L. Fejes Toth has drawn attention to the case when the n points consist of all points of a regular hexagonal lattice lying in the unit square, in which case the length of the shortest path is easily seen to be asymptotically equal to