8 research outputs found

    On imbalances in oriented multipartite graphs

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    Abstract. An oriented k-partite graph(multipartite graph) is the result of assigning a direction to each edge of a simple k-partite graph. Let D(V 1 , V 2 , Β· Β· Β· , V k ) be an oriented k-partite graph, and let d as the imbalance of the vertex v ij . In this paper, we characterize the imbalances of oriented k-partite graphs and give a constructive and existence criteria for sequences of integers to be the imbalances of some oriented k-partite graph. Also, we show the existence of an oriented k-partite graph with the given imbalance set

    On Scores, Losing Scores and Total Scores in Hypertournaments

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    A kk-hypertournament is a complete kk-hypergraph with each kk-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a kk-hypertournament, the score sis_{i} (losing score rir_{i}) of a vertex viv_{i} is the number of arcs containing viv_{i} in which viv_{i} is not the last element (in which viv_{i} is the last element). The total score of viv_{i} is defined as ti=siβˆ’rit_{i}=s_{i}-r_{i}. In this paper we obtain stronger inequalities for the quantities βˆ‘i∈Iri\sum_{i\in I}r_{i}, βˆ‘i∈Isi\sum_{i\in I}s_{i} and βˆ‘i∈Iti\sum_{i\in I}t_{i}, where IβŠ†{1,2,…,n}I\subseteq \{ 1,2,\ldots,n\}. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong kk-hypertournaments

    On the activities of p-basis of matroid perspectives

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    There is a renewed interest in matroid perspectives, either for their relevance in other fields of combinatorics and topology, or their applications in engineering. But, like for most of the Tutte invariants, computing the Tutte polynomial of matroid perspectives is #P-hard. Hence the importance of results whose applications would help to speed up computations. In the present paper, we show that a pseudobasis of a matroid perspective can be decomposed by a cyclic flat into two subsets, one of which has zero internal activity and the other has zero external activity. Apart from its own interest in understanding the internal structures of matroid perspective, this decomposition allows an expansion of the Tutte polynomial of matroid perspective over cyclic flats. This can be used to speed up the computation of various evaluations of the polynomial

    Cyclic flats and corners of the linking polynomial

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    Let T(M; x,y) = βˆ‘ij Tij xiyjdenote the Tutte polynomial of the matroid M. If Tij is a corner of T (M; x, y), then Tij counts the sets of corank i and nullity j and each such set is a cyclic flat of M. The main result of this article consists of extending the definition of cyclic flats to a pair of matroids and proving that the corners of the linking polynomial give the lower bound of the number of the cyclic flats of the matroid pair

    Sampling contingency tables

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    Let Ξ© be the set of all m Γ— n matrices, where r i and c j are the sums of entries in row i and column j, respectively. Sampling efficiently uniformly at random elements of Ξ© is a problem with interesting applications in Combinatorics and Statistics. To calibrate the statistic Ο‡ 2 for testing independence, Diaconis and Gangolli proposed a Markov chain on Ξ© that samples uniformly at random contingency tables of fixed row and column sums. Although the scheme works well for practical purposes, no formal proof is available on its rate of convergence. By using a canonical path argument, we prove that this Markov chain is fast mixing and the mixing time Ο„ x ( Ο΅ ) is given by Ο„ x ( Ο΅ ) ≀ 2 c m a x ( m n ) 4 ln ( c m a x Ο΅ βˆ’ 1 ) ,where c m a x βˆ’ 1 is the maximal value in a cell. Keywords: Sampling, Canonical path, Mixing time of Markov chain

    Rejection sampling of bipartite graphs with given degree sequence

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    Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where m=12βˆ‘i=1nai$m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999)
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