Abstract
It is proved that there are constants c1, c2, and c3 such that for any set S of n points in the unit square and for any minimum-lengths of T of S (1) the sum of squares of the edge lengths of T is bounded by c1 log n, (2) the sum of edge lengths of any subset E of T is bounded by c2|E|1/2, and (3) the number of edges having length t or greater in T is at most c3/t2. The second and third bounds are independent of the number of points in S, as well as their locations. Extensions to dimensions d>2 are also sketched.The presence of the logarithmic term in (1) is engaging because such a term is not needed in the case of the minimum spanning tree and several analogous problems, and, furthermore, we know that there always exists some tour of S (which perhaps does not have minimal length) for which the sum of squared edges is bounded independently of n.
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