The purpose of this note is to present a proof of Hall's matching theorem (also called marriage theorem) which I have not encountered elsewhere in literature-what yet does not mean that it is necessarily new. We refer to Hall's theorem in the following form: Theorem 1 (Hall). Let n be a positive integer. Let be a bipartite graph whose set of vertices consists of n blue vertices B 1 ; B 2 Some n...

Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a short program p transforming a to b that has the least possible length and is simple conditional on b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be general...

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those indu...

The matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex...

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing “deficiency” with multiplicity of θ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, whi...

Nominal techniques were introduced to represent in a simple and natural way systems that involve binders. The syntax includes an abstraction operator and a primitive notion of name swapping. Nominal matching is matching modulo α-equality, and has applications in programming languages and theorem proving, amongst others. In this paper we describe efficient algorithms to check the validity of equ...

A folk theorem for repeated matching games is established that holds if the stage game is not a pure coordination game. It holds independent of population size and for all matching rules—including rules that depend on players choices or the history of play. This paper also establishes an equilibrium condition and using this discovers two di¤erences between the equilibria of repeated matching ga...

A simple 2-matching in a graph is a subgraph all of whose nodes have degree 1 or 2. A simple 2-matching is called k-restricted if every connected component has > k edges. We consider the problem of finding a k-restricted simple 2-matching with a maximum number of edges, which is a relaxation of the problem of finding a Hamilton cycle in a graph. Our main result is a min-max theorem for the maxi...

Difference matching is a generalization of first-order matching where terms are made identical both by variable instantiation and by structure hiding. After matching, the hidden structure may be removed by a type of controlled rewriting, called rippling, that leaves the rest of the term unaltered. Rippling has proved highly successful in inductive theorem proving. Difference matching allows us ...