Embedding Products of Graphs into Euclidean Spaces

For any collection of graphs G1, . . . , GN we find the minimal dimension d such that the product G1 × · · · ×GN is embeddable into R . In particular, we prove that (K5) and (K3,3) are not embeddable into R, where K5 and K3,3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding LkO → S, where O is a vertex of (K5), has a pair of linked (n − 1)-spheres. Introduction. Our main result is the solution of the Menger problem from [Men29]: we prove that (K5) N 6 →֒ R . Moreover, for a given collection of graphs G1, . . . , GN we find the minimal dimension d such that G1 × · · · × GN →֒ R. We denote by Kn a complete graph on n vertices and by Kn,n a complete bipartite graph on 2n vertices. We write K →֒ R, if a polyhedron K is piecewise linearly embeddable into R. The topological problem of embeddability is a very essential one (e. g., see [Sch84, ReSk99, ARS01, Sko07]). Our special case of the problem is interesting because the complete answer can be obtained and is stated easily, but the proof is non-trivial and contains interesting ideas. Theorem 1. Let G1, . . . Gn be connected graphs, distinct from point, I and S . The minimal dimension such that G1 × · · · ×Gn × (S) × I →֒ R is d = { 2n+ s+ i, if either i 6= 0 or some Gk is planar (i. e., Gk 6⊃ K5,K3,3 ), (1) 2n+ s+ 1, otherwise. (2) Theorem 1 remains true in topological category. We first prove Theorem 1 in piecewise linear category and then deduce the topological version from the piecewise linear one. From now and till that moment we work in the piecewise linear category. Theorem 1 was stated (without proof) in [Gal93], cf. [Gal92]. The proof of embeddability is trivial (see below). The non-embeddability has been proved earlier in some specific cases. For example, it was known that Y n 6 →֒ R, where Y is a triod (letter ”Y”). A nice proof of this folklore result is presented in [Sko07], cf. [ReSk01]. Also it was known that K5 × S 6 →֒ R (Tom Tucker, private communication). In [Um78] it is proved that K5 × K5 6 →֒ R; that proof contains about 10 pages of calculations involving spectral sequences. We obtain a shorter geometric proof of this result (see Example 2 and Lemma 2 below). The proof of the non-embeddability in case (2), namely, Lemma 2, is the main point of Theorem 1 (while case (1) is reduced easily to a result of van Kampen.) Our proof of Theorem 1 is quite elementary, in particular, we do not use any abstract algebraic topology. We use a reduction to a problem from so-called Ramsey link theory [S81, CG83, SeSp92, RST93, RST95, LS98, Neg98, SSS98, T00, ShTa]. The classical Conway–Gordon–Sachs theorem of Ramsey link theory asserts that any embedding of K6 into R 3 has a pair of (homologically) linked cycles. In other words, K6 is not linklessly embeddable into R . The graph K4,4 has the same property (the Sachs theorem, proved in [S81]). Denote by σ n the m-skeleton of a n-simplex. For a polyhedron σ let σ be the join of n copies of σ. In our proof of Theorem 1 we use the following higher dimensional generalization of the Sachs theorem: 1991 Mathematics Subject Classification. 57Q35, 57Q45.