Hall's Condition Research Articles

Finite matchability under the matroidal Hall's condition

Aharoni and Ziv conjectured that if M and N are finitary matroids on E, then a certain “Hall-like” condition is sufficient to guarantee the existence of an M-independent spanning set of N. We show that their condition ensures that every finite subset of E is N-spanned by an M-independent set.

A generalization of balanced tableaux and marriage problems with unique solutions

We consider families of finite sets that we call shellable and that have been characterized by Chang and by Hirst and Hughes as being the families of sets that admit unique solutions to Hall's marriage problem. In this paper, we introduce a natural generalization of Edelman and Greene's balanced tableaux that involves families of sets that satisfy Hall's marriage Condition and certain words in $[n]^m$, then prove that shellable families can be characterized by a strong existence condition relating to this generalization. As a consequence of this characterization, we show that the average number of such generalized tableaux is given by a generalization of the hook-length formula.

State Dependent Control of Closed Queueing Networks

We study the design of state dependent control for a closed queueing network model, inspired by shared transportation systems such as ridesharing. In particular, we focus on the design of assignment policies, wherein the platform can choose which supply unit to dispatch to meet an incoming customer request. The supply unit subsequently becomes available at the destination after dropping the customer. We consider the proportion of dropped demand in steady state as the performance measure. We propose a family of simple and explicit state dependent policies called Scaled MaxWeight (SMW) policies and prove that under the complete resource pooling (CRP) condition (analogous to a strict version of Hall's condition for bipartite matchings), any SMW policy induces an exponential decay of demand-dropping probability as the number of supply units scales to infinity. Furthermore, we show that there is an SMW policy that achieves the optimal exponent among all assignment policies, and analytically specify this policy in terms of the matrix of customer-request arrival rates. The optimal SMW policy protects structurally under-supplied locations.

Finding Perfect Matchings in Bipartite Hypergraphs

Haxell's condition [16] is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied.We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell's condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g. Set Packing.

Every Graph $G$ is Hall $\Delta(G)$-Extendible

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.

Baire measurable paradoxical decompositions via matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in R3 using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann–Day problem: if a is a nonamenable action of a group on a Polish space X by Borel automorphisms, then there is a free Baire measurable action of F2 on X which is Lipschitz with respect to a.

Completing Partial Proper Colorings using Hall's Condition

In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph $G$ with list assignment $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $m$-completable for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable.

Hall's theorem and extending partial latinized rectangles

We generalize a theorem of M. Hall Jr., that an r×n Latin rectangle on n symbols can be extended to an n×n Latin square on the same n symbols. Let p, n, ν1,ν2,…,νn be positive integers such that 1≤νi≤p(1≤i≤n) and ∑i=1nνi=p2. Call an r×p matrix on n symbols σ1,σ2,…,σn an r×p(ν1,ν2,…,νn)-latinized rectangle if no symbol occurs more than once in any row or column, and if the symbol σi occurs at most νi times altogether (1≤i≤n). We give a necessary and sufficient condition for an r×p(ν1,ν2,…,νn)-latinized rectangle to be extendible to a p×p(ν1,ν2,…,νn)-latinized square. The condition is a generalization of P. Hall's condition for the existence of a system of distinct representatives, and will be called Hall's (ν1,ν2,…,νn)-Constrained Condition. We then use our main result to give two further sets of necessary and sufficient conditions. Finally we use our results to show that, given p, n, ν1,ν2,…,νn such that 1≤νi≤p, ∑i=1nνi=p2, then a p×p(ν1,ν2,…,νn)-latinized square exists.

TATA LETAK DAN KOMUNIKASI RUANG OUTLET VNC, SENAYAN CITY

This research aims to find out how the layout and space communications providing in VNC outlet Senayan City, And what message is being delivered to customers via VNC layout and space communications in the customer service outlets providing VNC, Senayan City. Theories used in research is the theory of Edward Hall, about the existence of intimate, personal, social distance is the distance, the distance of the public. According to Astrid Kusumowidagdo in his book the design of Retail (57: 2011) on the layout, i.e. Selling Space Space, Personnel, Merchandising Space, Customer Space. Because according to Shannon and Weaver (1949) that the communication is a form of interaction that interplay with each other, deliberate or accidental. Not tebatas in the form of communication using verbal language, but also in terms of expression, painting, art, and technology (nonverbal). The method of research is case studies with one case and multi unit of analysis. The collection of data obtained by the author with conducting research through participant observation in the natural situation of the research object of VNC, Senayan City outlets. Analysis of case studies on non-verbal communication when it is associated with this research is to describe How the layout and Space Communications Officer in the customer service Outlets Providing VNC, Senayan City The results showed a space communications outlets used VNC, Senayan City in providing services at the customer's personal distance communication, social distance, the distance of the public. With the consumer is not a mutual tangent with the officer and fellow visitors when dealing in the Outlet. The layout and space communication in the service of customers to note. The layout of individual items and accessories of other supporters have long distance interaction, where the customer can turned his attention to the products and services by making your employees satisfaction through comfort level optimal space. Harmony in structuring the space created when there is balance and alignment of the materials described above. The room must always provide a safe, healthy and peaceful, memorable. A summary of the results of this research is a communications space are good or comfortable reflected from the good and the room comfortable. Theoretical advice on non-verbal communication through layout and communications space spatial expressed very ease understanding of Edward Hall's condition through the distance and cue-cue. Practical advice so that the public understand the layout and space in daily communication in interacting with behaves the opposite communication. The layout of individual items and accessories of other supporters have long distance interaction, where the customer can turned his attention to the products and service by the employee so as to make optimal satisfaction through comfort level that was formed through the layout and communications space.

Relaxations of Hall’s Condition: Optimal batch codes with multiple queries

Combinatorial batch codes model the storage of a database on a given number of servers such that any k or fewer items can be retrieved by reading at most t items from each server. A combinatorial batch code with parameters n; k; m; t can be represented by a system F of n (not necessarily distinct) sets over an m-element underlying set X, such that for any k or fewer members of F there exists a system of representatives in which each element of X occurs with multiplicity at most t. The main purpose is to determine the minimum N(n; k; m; t) of total data storage ?F?F |F| over all combinatorial batch codes F with given parameters. Previous papers concentrated on the case t = 1. Here we obtain the first nontrivial results on combinatorial batch codes with t > 1. We determine N(n; k; m; t) for all cases with k ? 3t, and also for all cases where n ? t? m dk=te?2?. Our results can be considered equivalently as minimum total size ?F?F |F| over all set systems F of given order m and size n, which satisfy a relaxed version of Hall's Condition; that is, |UF?| ? |F?|/t holds for every subsystem F? ? F of size at most k.

Cropper's question and Cruse's theorem about partial Latin squares

In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ1,σ2,…,σt, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ1,σ2,…,σn, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σt + 1,σt + 2,…,σn}. These conditions consisted of r + s + t + 1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be sufficient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r + s + t + 1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:268-279, 2011

List multicoloring problems involving the k‐fold Hall numbers

AbstractWe show that the four‐cycle has a k‐fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ⌈5k/3⌉; furthermore, the same is not true with shorter list lengths. In terms of h(k)(G), the k ‐fold Hall number of a graph G, this result is stated as h(k)(C4)=2k−⌊k/3⌋. For longer cycles it is known that h(k)(Cn)=2k, for n odd, and 2k−⌊k/(n−1)⌋≤h(k)(Cn)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C4). We prove that if G is the diamond (a four‐cycle with a diagonal), then h(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.

HALL'S CONDITION AND IDEMPOTENT RANK OF IDEALS OF ENDOMORPHISM MONOIDS

AbstractIn 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

The k-fold list coloring of cycles with Hall's condition

We prove that any cycle Cn, n ≥ 4, with list assignment L, has a k -fold list coloring from the given lists if (i) each list contains at least 2k colors and (ii) Cn and L satisfy Hall's condition for k -fold list colorings. Further, 2k in (i) cannot be replaced by 2k - 1 if either n is odd, or n is even and n ≥ k + 2. In other words, if n ≥ 4, the k -fold Hall number of a cycle Cn satifies h(k)(Cn) ≤ 2k, with equality if n is odd, or n is even and n ≥ k + 2.

Characterization of graphs with hall number 2

AbstractHall's condition is a simple requirement that a graph G and list assignment L must satisfy if G is to have a proper L‐colouring. The Hall number of G is the smallest integer m such that whenever the lists on the vertices each has size at least m and Hall's condition is satisfied a proper L‐colouring exists. Hilton and P.D. Johnson introduced the parameter and showed that a graph has Hall number 1 if and only if every block is a clique. In this paper we give a forbidden‐induced‐subgraph characterization of graphs with Hall number 2. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 81–100, 2004

Hall parameters of complete and complete bipartite graphs

AbstractGiven a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well‐known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ϕ with ϕ(υ) ∈ L (υ) (∀υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non‐triviality condition, called Hall's condition, has to be satisfied by the list assignment. The edge‐analogue of the Hall number is called the Hall index, h′(G), and the total analogue is called the total Hall number, h″(G), of G. If the stock of colours from which L(υ) is selected is restricted to a set of size k, then the analogous numbers are called k‐restricted, or restricted, Hall parameters, and are denoted by hk(G), h′k(G) and h″k(G).Our main object in this article is to determine, or closely bound, h′(K), h″(Kn), h′(Km,n) and h′k(Km,n). We also answer some hitherto unresolved questions about Hall parameters. We show in particular that there are examples of graphs G with h′(G)−h′(G − e)>1. We show that there are examples of graphs G and induced subgraphs H with hk(G)<hk(H) [this phenomenon cannot occur with unrestricted Hall numbers]. We also give an example of a graph G and an integer k such that hk(G)<χ(G)<h(G). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 208–237, 2002

Restricted list coloring and Hall's condition

If L is a list assignment of colors to the vertices of a graph G of chromatic number χ( G), a certain condition on L and G, known as Hall's condition, which is obviously necessary for G to have an L-coloring, is known to be sufficient if and only if each block of G is a clique. We show that if the set of colors from which the lists are drawn has size χ( G) then there exist graphs G for which Hall's condition is sufficient for an L-coloring even though not every block of G is a clique. But if the set of colors has size greater than χ( G), then Hall's condition is again sufficient if and only if each block of G is a clique.

Hall's condition for list-coloring, and the Hall parameters: recent developments

Hall's condition for list-coloring, and the Hall parameters: recent developments

Combinatorial properties of the family of maximum stable sets of a graph

The stability number α ( G ) of a graph G is the size of a maximum stable set of G , core (G)=⋂{S : S is a maximum stable set in G }, and ξ ( G )=| core ( G )|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ ( G )⩽1, then G is quasi-regularizable; (ii) if the order of G is n , and α ( G )>( n + k −min{1,| N ( core ( G ))|})/2, for some k ⩾1, then ξ ( G )⩾ k +1; moreover, if n + k −min{1,| N ( core ( G ))|} is even, then ξ ( G )⩾ k +2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ ( G )⩾1 is true whenever α ( G )> n /2. In the case of König–Egerváry graphs, i.e., for graphs enjoying the equality α ( G )+ μ ( G )= n , where μ ( G ) is the maximum size of a matching of G , we prove that | core ( G )|>| N ( core ( G ))| is a necessary and sufficient condition for α ( G )> n /2. Furthermore, for bipartite graphs without isolated vertices, ξ ( G )⩾2 is equivalent to α ( G )> n /2. We also show that Hall's Marriage Theorem is true for König–Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core ( G ).

Relations among the fractional chromatic, choice, Hall, and Hall-condition numbers of simple graphs

Hall's condition for the existence of a proper vertex list-multicoloring of a simple graph G has recently been used to define the fractional Hall and Hall-condition numbers of G, h f(G) and s f(G) . Little is known about h f(G) , but it is known that s f(G)= max[|V(H)|/α(H);H⩽G] , where ‘ ⩽’ means ‘is a subgraph of’ and α(H) denotes the vertex independence number of H. Let χ f(G) and c f(G) denote the fractional chromatic and choice (list-chromatic) numbers of G. (Actually, Slivnik has shown that these are equal, but we will continue to distinguish notationally between them.) We give various relations among χ f(G) , c f(G) , h f(G) , and s f(G) , mostly notably that χ f(G)=c f(G)=s f(G) , when G is a line graph. We give examples to show that this equality does not necessarily hold when G is not a line graph. Relations among and behavior of the ‘ k-fold’ parameters that appear in the definitions of the fractional parameters are also investigated. The k-fold Hall numbers of the claw are determined and from this certain conclusions follow—for instance, that the sequence (h (k)(G)) of k-fold Hall numbers of a graph G is not necessarily subadditive.