© 1998 by London Mathematical Society
© The London Mathematical Society
An Application of Spanning Trees to k-Point Separating Families of Functions
Department of Mathematics, University of South Florida Tampa, FL 33620-5700, USA. E-mail: eclark{at}math.usf.edu
Department of Mathematics, University of South Florida Tampa, FL 33620-5700, USA. E-mail: mccolm{at}math.usf.edu
Department of Mathematics, University of South Florida Tampa, FL 33620-5700, USA. E-mail: boris{at}math.usf.edu
Received 14 February 1993. Revision received 5 September 1995.
A family F of functions from Rn to R is k-point separating if, for every k-subset S of Rn, there is a function f
F such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family F has cardinality n(k–1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on Rn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k
2 vertices and n(k–1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k–1) edges to be an n-tree.