For the assignment game, we analyze the following mechanism: sellers, simultaneously, fix their prices first; then buyers, sequentially, decide which object to buy, if any, among the remaining objects. The first phase of the game determines the potential prices, while the second phase determines the actual matching. We prove that the set of subgame perfect equilibria in pure strategies in the s...

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) [Formula: see text] of a tree T of given order. While the trees that attain the lower bound are easily characterised, the trees with the largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of ...

It is conjectured (and proved for edge sets of graphs) in [1] that in every family d of finite sets a subfamily ~ of disjoint sets (called a 'strongly maximal matching') exists, so that no replacement of k of them by more than k sets from a / resu l t s again in a subfamily of disjoint sets. As expected by Erd6s (Introduction of [2]), the conjecture is false. A counterexample is ag, the family ...

The following properties of the core of a one-to-one matching problem are well-known: (i) the core is non-empty; (ii) the core is a lattice; and (iii) the set of unmatched agents is identical for any two matchings belonging to the core. The literature on two-sided matching focuses almost exclusively on the core and studies extensively its properties. Our main result is the following characteriz...

Let G = (S, T, E) be a bipartite-graph, where S U T is the set of nodes (S n T = 8) and E is the set of edges, E c S X T. Let S = {ur , . . . . II,}, T = {VI, . . . . vt} (A t), and 1 E I= e. An (S, T) matching is a subset M of E such that no two edges in M have a common endpoint. A maximum matching is a matching of maximum cardinality. The set of nodes which take part in such a maximum matchin...

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343–358) introduced a graph operator D(G), called the D-graph of G, which has been useful in investigating the structural aspects of maximal Tutte sets in G with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph D(G) and maximal extreme sets in G. Th...

Given an assignment of weights w to the edges of an infinite graph G, a matching M in G is called strongly w-maximal if for any matching N there holds ∑ {w(e) | e ∈ N \ M} ≤ ∑ {w(e) | e ∈ M \ N}. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching. This result is best possible in the sense that if we allow irrational values or ...

Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For this reason, there exist many recent randomized results that aim provide guarantee expected. The strongest possible algorithm is it always correct (Las Vegas) and high...