Basic Embeddings Into A Product of Trees
by
V. Kurlin
Department of High Geometry and Topology, Moscow State University, Russia

The notion of basic embedding appeared in research motivated by Kolmogorov-Arnold's solution of Hilbert's 13th problem. Let K, X, Y be topological spaces. An embedding K subset X ×Y is called basic if for every continuous function f : K --> R there exist continuous functions g : X --> R, h : Y --> R such that f(x, y) = g(x) + h(y) for any point (x, y) in K.

Theorem. A finite (not necessary connected) graph K is basically embeddable into T_m ×T_n if and only if it is a tree and either \delta(K) < m+n-2 or \delta(K) = m+n-2 and K has a horrid vertex with a hanging edge.

Let T_i be an i-od. We call a vertex of K horrid (resp. awful) if its degree is greater than 4 (resp its degree equals 4 and it has no hanging edges). The defect of graph K is the sum \delta(K) = (degA₁ - 2) + ... + (degA_k - 2), where A₁, ... , A_k are all horrid and awful vertices of K. Our theorem is a generalisation of Skopenkov's description of graphs, basically embeddable into R² and our proof is a (non-trivial) extension of that one.

Date received: June 25, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-32.