132 research outputs found
Pairing games and markets
Pairing Games or Markets studied here are the non-two-sided NTU generalization of assignment games. We show that the Equilibrium Set is nonempty, that it is the set of stable allocations or the set of semistable allocations, and that it has has several notable structural properties. We also introduce the solution concept of pseudostable allocations and show that they are in the Demand Bargaining Set. We give a dynamic Market Procedure that reaches the Equilibrium Set in a bounded number of steps. We use elementary tools of graph theory and a representation theorem obtained here
Gallai-Edmonds decomposition of unicyclic graphs from null space
In this paper, we compute the Gallai-Edmonds decomposition of a unicyclic graph G using linear algebraic tools. More precisely, the Gallai-Edmonds decomposition of G is obtained from the null space associated with adjacency matrices of its subtrees
An improvement on Łuczak's connected matchings method
A connected matching in a graph G is a matching contained in a connected component of G. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007)
Monochromatic connected matchings in 2-edge-colored multipartite graphs
A matching in a graph is connected if all the edges of are in the
same component of . Following \L uczak,there have been many results using
the existence of large connected matchings in cluster graphs with respect to
regular partitions of large graphs to show the existence of long paths and
other structures in these graphs. We prove exact
Ramsey-type bounds on the sizes of monochromatic connected matchings in
-edge-colored multipartite graphs. In addition, we prove a stability theorem
for such matchings.Comment: 29 pages, 2 figure
- …