A necessary and sufficient condition for the existence of a complete stable matching
Journal of Algorithms
Fractional kernals in digraphs
Journal of Combinatorial Theory Series B
A fixed-point approach to stable matchings and some applications
Mathematics of Operations Research
A fractional model of the border gateway protocol (BGP)
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The stable roommates problem with choice functions
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
SIAM Journal on Discrete Mathematics
Note: An algorithm for a super-stable roommates problem
Theoretical Computer Science
The integral stable allocation problem on graphs
Discrete Optimization
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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 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.