Kernels, Stable Matchings, and Scarf's Lemma

Scarf's Lemma originally appeared as a tool to prove the non-emptiness of the core of certain NTU games. More recently, however, several applications have been found in the area of graph theory and discrete mathematics. In this paper we present and extend some of these applications. In particular, we prove results on the existence of kernels in orientations of h-perfect graphs. We describe a new direct link between Scarf's Lemma and Sperner's Lemma giving a new proof to the former.

[1]  Ali Ridha Mahjoub,et al.  Compositions of Graphs and Polyhedra III: Graphs with No W4 Minor , 1994, SIAM J. Discret. Math..

[2]  Jimmy J. M. Tan.  A Necessary and Sufficient Condition for the Existence of a Complete Stable Matching , 1991, J. Algorithms.

[3]  Vladimir Gurvich,et al.  Perfect graphs, kernels, and cores of cooperative games , 2006, Discret. Math..

[4]  Tamás Fleiner,et al.  On a lemma of Scarf , 2002, J. Comb. Theory, Ser. B.

[5]  H. Scarf.  The Core of an N Person Game , 1967 .

[6]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[7]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computations , 1965 .

[8]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[9]  Michel Balinski,et al.  Integer Programming: Methods, Uses, Computation , 2010, 50 Years of Integer Programming.

[10]  Robert W. Irving.  An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[11]  C. Rioux.  Scarf's Theorem and Applications in Combinatorics , 2006 .

[12]  Vladimir Gurvich,et al.  Perfect graphs are kernel solvable , 1996, Discret. Math..

[13]  V. Gurvich,et al.  Stable families of coalitions and normal hypergraphs , 1997 .

[14]  J. P. Uhry,et al.  A class of h-perfect graphs , 1984, Discret. Math..

[15]  F. Bruce Shepherd,et al.  Applying Lehman's theorems to packing problems , 1995, Math. Program..

[16]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[17]  Ron Holzman,et al.  Fractional Kernels in Digraphs , 1998, J. Comb. Theory, Ser. B.

[18]  George L. Nemhauser,et al.  Polyhedral Characterizations, Perfection of Line Graphs , 1998, Discret. Appl. Math..