Case 1: Choose any x ∈ VI and any y ∈ ∂{x} (by hypothesis, ∂{x} has at least two elements). Let G∗ be the bipartite graph with input set V ∗ I = VI − {x}, output set V ∗ O = VO − {y}, and whose edges are the same as those of G, but with edges incident to either x or y deleted. The bipartite graph G∗ satisfies the hypothesis (1.1), because in Case 1 every proper subsetA ⊂ VI has |∂A| ≥ |A|+1, so...

We study a strategic model of wage negotiations between …rms and workers. First, we de…ne the stability of an allocation in an environment where …rms can employ more than one worker. Secondly, we develop a one-to-many non-cooperative matching game, which is an extension of Kamecke’s one-to-one non-cooperative matching game. The main result shows that the equivalence between the stable allocatio...

In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W (1)2 + 2W (1) ≈ 1.456, where W is the Lambert W -functio...

The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size boun...

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short PMH-property) if each of its matchings can be extended to Hamiltonian cycle. In this paper we establish some sufficient conditions for $G$ in order guarantee that line $L(G)$ PMH-property. particular, prove happens when is (i) with maximum degree at most $3$, (ii) complete graph, or (iii) an arbitraril...

The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a xed bi-partition of the vertices, there is no perfect matching between them. Therefore , it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size boun...

In a bipartite graph G, a set S ⊆ V (G) is deficient if |N(S)| < |S|. A matching M (with vertex set U) is k-suitable if G − U has no deficient set of size less than k. Let fk(d) be the maximum r such that in the d-dimensional hypercube Qd every k-suitable matching having size at most r extends to a perfect matching. We generalize results of Limaye and Sarvate by proving that fk(d) = k(d− k) + (...

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.