Published in:
Journal of Combinatorial Theory B 87 (2003) 72-80
The aim of this note is to bring closer to the center of the combinatorial scene a lemma of Scarf. The usefulness of the lemma in combinatorics has already been demonstrated in \cite{aharoniholzman}, where it was used to prove the existence of fractional kernels in digraphs not containing cyclic triangles. We provide coordinates for the lemma, both in terms of the family of combinatorial results it belongs to (that is, of the Gale-Shapley theorem) and in terms of its proof (identifying it as a kin of Sperner's lemma). We use the lemma to prove a fractional version of the Gale-Shapley theorem for hypergraphs, which in turn directly implies an extension of this theorem to general (not necessarily bipartite) graphs due to Tan. We also prove the following result, related to a theorem of Sands, Sauer and Woodrow: given a family of partial orders on the same ground set, there exists a system of weights on the vertices, which is (fractionally) independent in all orders, and each vertex is dominated by them in one of the orders.
Keywords:
Stable matching, Partial order, Matroid, Simplicial complex
Bibtex entry:
@techreport{egres-01-15,
AUTHOR | = | {Aharoni, Ron and Fleiner, Tam{\'a}s}, |
TITLE | = | {On a lemma of Scarf}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2001}, |
NUMBER | = | {TR-2001-15} |