A necessary condition on minimal cube numberings

Abstract

Let G = (V, E) be a graph with N vertices and A a set of N real numbers. Then a one-to-one mapping ϕ: V → A is called a numbering of G. The elements of A will always be assumed ordered a1 ≦ a2 ≦ … ≦ aN. In [3] and [7] it was shown how to construct numberings of the n-cube, in fact all numberings, which minimize Σe ∈ E Δe where ∆e = |ϕ(v) – ϕ(w)| and e is the edge between vertices v and w. A variant problem of considerable interest is to do the same for Σe ∈ E (Δe)2. It is conjectured that when A = {1,2, …,2n} the natural numbering is the unique minimizer of Σe ∈ E (Δe)2 for every n, but this has so far resisted all efforts. Theorem 1 in the following is the result of attempts to get weaker results, namely to thin out the ranks of those numberings which could possibly minimize Σe ∈ E (Δe)2. The second problem posed and solved in this paper is a generalization of the results in [3], where the n-cube becomes the n-torus.