On the connectedness of some geometric graphs

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Abstract

Fos SR, let G(Sd) denote the graph, obtained from all the points of the real d-space Rd having their coordinates in S, by connecting every two points which are at distance one. If S = Q (the rationals), or Q[√k], or Q[a] where a is a real algebraic number, then G(Sd) is shown to be connected for all large values of d. Similar results are obtained in the case of S = Q[nk], where the distance is computed in the lp-norm, p an integer, p ≥ 3 and np (or n = 2).

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Research supported in part by Natural Sciences and Engineering Research Council, Canada.