相关论文: Hall's marriage theorem
Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory,…
Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two…
The paper is devoted to a proof of the de Bruijn-Erd\"os theorem in incidence geometry based on the Ph. Hall's marriage theorem (the theorem about the systems of distinct representatives).
We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i…
We present a generalization of the marriage problem underlying Hall's famous Marriage Theorem to what we call the Symmetric Marriage Problem, a problem that can be thought of as a special case of Maximal Weighted Bipartite Matching. We show…
We consider a set-valued mapping on a simple graph and ask for the existence of a disparate selection. The term disparate is defined in the paper and we present a sufficient and necessary condition for the existence of a disparate…
We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of…
In this paper, we mainly investigate the converse of a well-known theorem proved by P. Hall, and present detailed characterizations under the various assumptions of the existence of some families of Hall subgroups. In particular, we prove…
This paper is concerned with unreachable pawn diagrams and the subset of which can be generated using Hall's Marriage Theorem. The result is 1 in 23 diagrams are unreachable by applying the theorem.
We generalize Philip Hall's celebrated theorems on finite solvable groups to scheme theory. Our result is based on a series of results on hypergroups.
Holroyd and Propp used Hall's marriage theorem to show that, given a probability distribution pi on a finite set S, there exists an infinite sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the number of i in…
We present a fascinating model that has lately caught attention among physicists working in complexity related fields. Though it originated from mathematics and later from economics, the model is very enlightening in many aspects that we…
In 1962, Gale and Shapley \cite{GS} introduced the concept of stable marriages and proved their existence. Since then, the statement of the stability problem has been highly generalized. And a lot of proofs has emerged for the existence in…
We propose a generalization of Halls marriage theorem. The generalization given here provides a necessary sufficient condition for arranging a successful friendship among n number of k sets. We define multimatrix, multideterminant,…
In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve…
In this paper, we establish the theory of $\sigma$-solvable hypergroups, study some properties of $\sigma$-solvable hypergroups and give similar results of Hall's Theorem in $\sigma$-solvable hypergroups.
Given a collection $\C$ of subsets of a finite set $X$, let $\bigcup \C = \cup_{S \in \C}S$. Philip Hall's celebrated theorem \cite{hall} concerning `systems of distinct representatives' tells us that for any collection $\C$ of subsets of…
In 1922, JBS Haldane discovered an intriguing bias of postzygotic isolation during early speciation: the heterogametic sex of F1 hybrids between closely related species or subspecies is more susceptible to sterility or inviability than the…
We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant,…