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JCT

1998

1998

The aim of this note is to point out some combinatorial applications of a lemma of Scarf, proved ﬁrst in the context of game theory. The usefulness of the lemma in combinatorics has already been demonstrated in a paper by the ﬁrst author and R. Holzman (J. Combin. Theory Ser. B 73 (1) (1998) 1) where it was used to prove the existence of fractional kernels in digraphs not containing cyclic triangles. We indicate some links of the lemma to other combinatorial results, both in terms of its statement (being a relative of the Gale–Shapley theorem) and its proof (in which respect it is 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 (J. Algorithms 12 (1) (1991) 154). We also prove the following result, related to a theorem of Sands et al. (J. Combin. Theory Ser. B 33 (3) (1982) 271): given a fam...

Related Content

Added |
22 Dec 2010 |

Updated |
22 Dec 2010 |

Type |
Journal |

Year |
1998 |

Where |
JCT |

Authors |
Ron Aharoni, Ron Holzman |

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