8 research outputs found
On imbalances in oriented multipartite graphs
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
A -hypertournament is a complete -hypergraph with each -edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a -hypertournament, the score (losing score ) of a vertex is the number of arcs containing in which is not the last element (in which is the last element). The total score of is defined as . In this paper we obtain stronger inequalities for the quantities , and , where . Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong -hypertournaments
On the activities of p-basis of matroid perspectives
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
Let T(M; x,y) = βij 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
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΅ ) β€ 2 c m a x ( m n ) 4 ln ( c m a x Ο΅ β 1 ) ,where c m a x β 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
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 β(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=12βi=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)