P. Hall. On representatives of subsets. J. London Math. Soc., 10:26--30, 1935.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with the variables. In such a matching, no edges are adjacent and all variable nodes are assigned to a value node, which satisfies the restriction that all variables must be pairwise di#erent. Necessary and su#cient conditions for the existence of such a matching were first given by Hall [Hal35] Theorem 5.1 Theorem (V , E) be a bipartite graph with partition (X, Y ) Let N(S) denote the set of neighbours of S V . Then G contains a matching that saturates every node in X if and only if for all S X (5.1) A global check for the satisfiability of the Alldifferent constraint ....

P. Hall. On representatives of subsets. J. London Math. Soc., 10:26-- 30, 1935.

....(s, sin) of m distinct elements such that si Si. For each i, si is called a representative of Si. For example, the set 1, 3, 2) is a SDR for the sets S = 1 , S2 = 1, 3 and Sa = 2, 3 . In 1935, P. Hall proved the following theorem for SRDs. The proof of this result can be found in [13]. THEOREM I (P. Hall, 1935, 13] Let S1, S2, Sm be a collection of m finite sets. Then an DR for these sets exists if and only if, for all k (0, 1, m , Sil tO Si2 to. to Sik] k, where the k sets Si, Sik represent any collection of k sets chosen from the m sets S, S, Sin. ....

....elements such that si Si. For each i, si is called a representative of Si. For example, the set 1, 3, 2) is a SDR for the sets S = 1 , S2 = 1, 3 and Sa = 2, 3 . In 1935, P. Hall proved the following theorem for SRDs. The proof of this result can be found in [13] THEOREM I (P. Hall, 1935, [13]) Let S1, S2, Sm be a collection of m finite sets. Then an DR for these sets exists if and only if, for all k (0, 1, m , Sil tO Si2 to. to Sik] k, where the k sets Si, Sik represent any collection of k sets chosen from the m sets S, S, Sin. The above theorem states that a ....

P. Hall, On representative of subsets, J. London Math. Soc., 10, 1935, 26-30.

....balanced if and only if it is a bipartite graph. Balanced hypergraphs can be characterized by a bicoloring theorem [2] The nodes of a balanced hypergraph can be colored either red or blue in suchaway that every edge with at least two nodes contains both a red node and a blue node. Hall s theorem [26] about the existence of a perfect matching in a bipartite graph extends to balanced hypergraphs [14] Further results on balanced matrices are surveyed in [12] Totally unimodular matrices A matrix is totally unimodular if every square submatrix has a determinant equal to 0, 1 or 1. A ....

P. Hall, On representatives of subsets, J. London Math. Soc. (1935) 26-30.

....storage node j, the constructed graph is an m regular bipartite graph. For the first time slot, whether there exists a set of requests each of which accesses a different storage node from a different delivery node is equivalent to find a perfect matching in G. According to the marriage theorem [12], if G is a k regular bipartite graph with k 0, then G has a perfect matching. After determining a perfect matching for time slot 0, eliminate the matched edges, the original problem of scheduling Nm requests to m time slots is reduced to a problem of scheduling N (m Gamma 1) requests to (m ....

P. Hall. On representatives of subsets. J. London Math. Soc., 10:26--30, 1935.

....augmenting graphs RRR 46 2001 Page 7 From now on, we will only consider minimal augmenting (C 4 ; S 1;2;4 ) free graphs H = W; B; E) which contain at least one black vertex b of degree 3. We denote x 1 ; x 2 ; x k (k 3) the white neighbors of b. From Lemma 1 and Hall s Theorem [9], we know that there is a perfect matching, denoted M(b) in the subgraph of H induced by V (H) fbg. We denote B(b) fm(x 1 ) m(x 2 ) m(x k )g the set of black vertices such that m(x i ) is the vertex matched with x i in M(b) Notice that no x i is adjacent to a m(x j ) with i 6= j, ....

P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26-30.

....Jacobaeus method. 2.4 Rearrangeably Nonblocking Clos Networks The Slepian Duguid theorem [3] shows that a Clos network is rearrangeably nonblocking if m n . To prove this, it is sufficient to consider a maximal assignment with m n = This result relies on a combinatorial theorem due to Hall [4], which is often called Hall s marriage theorem for reasons that will become clear. It can be stated informally like this. Suppose there are a certain number of boys and the same number of girls, and it is necessary to pair each boy off with a girl that he knows in order to be married. Hall ....

P. Hall: "On Representatives of Subsets", Journal of the London Mathematical Society, vol. 10, 1935, pp26-30

....and all the orderings are pairwise disjoint. In particular, as long as all singletons in the collection are distinct, the elements in the first position of a disjoint ordering form a system of distinct representatives. So for a disjoint ordering to exist, the conditions in Hall s matching theorem [9] must be satisfied. The converse is also true. Theorem 3.1 (Gao et al. 1998) For any finite collection of nonempty finite sets in which all singletons are distinct, there is a disjoint ordering if and only if there is a system of distinctive representatives. Recall that a system of distinctive ....

P. Hall, "On representatives of subsets," J. London Math. Soc., vol. 10, 1935, pp 26--30.

.... V (G) Proof. For the proof, see Bondy and Murty [3, p. 54] Recall that a matching in a simple graph G is a subset of mutually vertex disjoint edges of G. A matching is perfect if every vertex in G is on some edge of the matching. The following is a consequence of a well known theorem of Hall [13]. INTERFERENCE MINIMIZING COLORINGS 29 Proposition 5.2 (Marriage Theorem) If G is a d regular bipartite graph with d 0, then G has a perfect matching. Proof. For the proof, see Bondy and Murty [3, p. 73] We study the function #(n, d; r) defined by #(n, d; r) # # # 1 if there exists an ....

P. Hall, On representatives of subsets, J. London Math. Soc., 10 (1935), pp. 26--30.

....shall show that jG n B n j jB n n Gj. Let us de ne a bipartite graph, the vertices of which are the sets V 1 = G n B n and V 2 = B n n G, and let us connect a vertex x 2 V 1 to a vertex y 2 V 2 if and only if x 2 U(y; B n ) Our claim, i.e. jV 1 j jV 2 j, will follow by Hall s theorem (see [5]) For this end, it is enough to show that for any subset A V 2 the set of neighbors of A (in V 1 ) is at least as large as A, i.e. that V 1 [ a2A U(a; B n ) jAj: 4.14) Let us note rst that for any nonempty subset A B n , the set D(A) 4 a2A U(a; B n ) is ....

Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), pp. 26-30.

....2 P , P p: pq (p; q) q) This is sometimes called a fractional matching, and sometimes a Markov kernel. Note that if P = hA [ B; OEi, where OE A Theta B (and A B = and if G is the integers and j(a) 1 for each a 2 A while (b) 1 for each b 2 B, then we get Hall s Marriage Theorem [Hal35]. If we let j and be any integral values, we get a theorem of Hoffman [Ho60] see [Re84] If we let G be the reals, we get the finite version of the theorem of Strassen [St65] KamKO77] or Major [LoP86, p. 76] If G is Q[ p 5] we get a funny beast indeed. Note that if we got Theorem 3.1 ....

P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26--30.

....Jacobaeus method. 2.4 Rearrangeably Nonblocking Clos Networks The Slepian Duguid theorem [3] shows that a Clos network is rearrangeably nonblocking if m n # . To prove this, it is sufficient to consider a maximal assignment with m n = This result relies on a combinatorial theorem due to Hall [4], which is often called Hall s marriage theorem for reasons that will become clear. It can be stated informally like this. Suppose there are a certain number of boys and the same number of girls, and it is necessary to pair each boy off with a girl that he knows in order to be married. Hall ....

P. Hall: "On Representatives of Subsets", Journal of the London Mathematical Society, vol. 10, 1935, pp26-30

....algorithm. In the last part of the paper, we will adapt a technique from [Knu73] to describe an alternative method, and show that it can be used to improve the Select ordering and check algorithm. We start with a simple technical lemma a corollary to the celebrated Hall s theorem [Hal35], see also [Bol78] Lemma 2.3 Let G = V [ U; E) be a bipartite graph with vertex classes V and U in which all the vertices in V have the same degree, and all the vertices in U have the same degree. If jU j jV j, then the maximum matching in G covers all vertices in V . Proof: Let us denote jV j ....

P. Hall. On representatives of subsets. J. of London Math. Soc., 10:26--30, 1935.

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P. Hall, On representatives of subsets, J. London Math. Soc., 10 (1935), 26-30.

....ordering form a system of distinct representatives. For example, the following are four sets and a disjoint ordering for them. Note that the initial elements of the ordering, i.e. 1, 3, 2, 4, form a system of distinct representatives for X 1 ,X 2 ,X 3 ,X 4 . A well known theorem of P. Hall [3], often called Hall s matching theorem, says that a family of finite sets has a system of distinct representatives (SDR) if and only if the union of any k sets contains at least k distinct elements. The condition in Hall s theorem is known as the marriage condition. Obviously the marriage ....

P. Hall, "On representatives of subsets," J. London Math. Soc., vol. 10, 1935, pp 26--30.

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P. Hall. On representatives of subsets. J. London Math. Soc., 10:26--30, 1935.

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P. Hall. On representatives of subsets. J. London Math. Soc., 10:26--30, 1935.

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Phillip Hall, "On representatives of subsets," J. London Math. Soc.,vol. 10, pp. 26--30, 1936.

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P. Hall, "On representatives of subsets," The Journal of the London Mathematical Society, 10, pp. 26--30, 1935.

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P. Hall (1935) On representatives of subsets. Journal of London Mathematical Society 10: 26-30.

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P. Hall, "On representatives of subsets," The Journal of the London Mathematical Society, 10, pp. 26--30, 1935.

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P. Hall, On representatives of subsets, J. London Math. Soc., 10 (1935), 26--30.

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Hall, P., On representatives of subsets, J. London Math. Soc. 10, (1935), pp. 26-30.

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P. Hall, "On representatives of subsets", J. London Math. Soc. 10, 1935, pp. 26-30.

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Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), pp. 26-30.

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P. Hall, "On representatives of subsets", J. London Math. Soc., 10, 1935, pp. 26-30. 28

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