On the Realization of Networks in Three-Dimensional Space

Abstract

By a (d, n)-network we shall mean a oriented graph with n numbered vertices α ₁, α ₂, ..., α _n and dn marked edges such that precisely d edges are incident to each vertex and one of them is marked by the weight x ₁, another by the weight x ₂, etc., and finally the last one by the weight x _d .

Problemy Kibernetiki, 1967, N19, p. 261–268.

At the beginning of the sixties A. N. Kolmogorov proved Theorem 1 for nets with bounded branching and also the following approximation to Theorem 2: there exists a net with n > 1 elements any realization of which has diameter greater than $C\sqrt n /\log n$, where C is a certain constant not depending on n. The final versions of the theorems given here belong to Ya. M. Barzdin. Author’s note.