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AN APPLICATION OF SPANNING TREES TO k-POINT SEPARATING FAMILIES OF FUNCTIONS

Published online by Cambridge University Press:  01 October 1998

W. EDWIN CLARK ,
GREGORY L. McCOLM  and
BORIS SHEKHTMAN
W. EDWIN CLARK
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA. E-mail: eclark@math.usf.edu; mccolm@math.usf.edu; boris@math.usf.edu
GREGORY L. McCOLM
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA. E-mail: eclark@math.usf.edu; mccolm@math.usf.edu; boris@math.usf.edu
BORIS SHEKHTMAN
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA. E-mail: eclark@math.usf.edu; mccolm@math.usf.edu; boris@math.usf.edu
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Abstract

A family [Fscr ] of functions from ℝn to ℝ is k-point separating if, for every k-subset S of ℝn, there is a function f∈[Fscr ] 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 [Fscr ] 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[ges ]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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 58 , Issue 2 , October 1998 , pp. 297 - 310
Copyright
The London Mathematical Society 1998

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