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Let represent the minimum number of complete -partite -graphs required to partition the edge set of the complete -uniform hypergraph on vertices. The Graham–Pollak theorem states that . An upper bound of was known. Recently this was improved to for even . A bound of was also proved recently. Let be the limit of as . The smallest odd for which that was known was for . In this note we improve this to and also give better upper bounds for , for small values of even . 相似文献
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The Hankel determinants of the convolution powers of Catalan numbers were considered by Cigler and Krattenthaler. We evaluate these determinants for by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin’s continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjecture a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants . Similar results are obtained. 相似文献
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《Discrete Mathematics》2020,343(6):111842
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We consider a -parameter Hermite process with Hurst index and we study its limit behavior in distribution when the Hurst parameters (or a part of them) converge to and/or 1. The limit obtained is Gaussian (when at least one parameter tends to ) and non-Gaussian (when at least one-parameter tends to 1 and none converges to ). 相似文献
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J. Wu 《Indagationes Mathematicae》2019,30(4):536-541
In this short note, we prove that for , where is the Euler totient function and is the integral part of real . This improves recent results of Bordellès–Heyman–Shparlinski and of Dai–Pan. 相似文献
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A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. We use to denote the 3-uniform hypergraph whose vertex set can be partitioned into two vertex classes and of size and , respectively, and whose edge set consists of all the triples containing at least two vertices of . Let be a 3-uniform hypergraph of order with no isolated vertex and for any two adjacent vertices . In this paper, we show that contains a matching of size if and only if is not a subgraph of . This result improves our previous one in Zhang and Lu (2018). 相似文献
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Let be a simple connected graph with vertices and edges. The spectral radius of is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of which improves some known bounds: If , where is an integer, then The equality holds if and only if is a complete graph or , where is the graph obtained from by deleting some edge . 相似文献
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《Discrete Mathematics》2023,346(1):113151
The Plurality problem - introduced by Aigner - has many variants. In this article we deal with the following version: suppose we are given n balls, each of them colored by one of three colors. A plurality ball is one such that its color class is strictly larger than any other color class. Questioner asks a triplet (or a k-set in general), and Adversary as an answer gives the partition of the triplet (or the k-set) into color classes. Questioner wants to find a plurality ball as soon as possible or show that there is no such ball, while Adversary wants to postpone this.We denote by the largest number of queries needed to ask in the worst case if both play optimally. We provide an almost precise result in the case of even n by proving that for even we have and for odd we haveWe also prove some bounds on the number of queries needed to ask in the case . 相似文献
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《Discrete Mathematics》2020,343(8):111922
Tribonacci cubes are induced subgraphs of , obtained by removing all the vertices that contain more than two consecutive 1’s. In the present work, we give some enumerative properties related to . We show that the number of vertices of weight in is and express the number of edges of these graphs in terms of convolved Tribonacci numbers. We investigate the cube polynomials of Tribonacci cubes and determine the corresponding generating function. Finally, we give a formula for the number of induced -cubes in . 相似文献
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《Discrete Mathematics》2020,343(10):111996
A Gallai coloring of a complete graph is an edge coloring without triangles colored with three different colors. A sequence of positive integers is an -sequence if . An -sequence is a G-sequence if there is a Gallai coloring of with colors such that there are edges of color for all . Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer there exists an integer such that every -sequence is a G-sequence if and only if . They showed that and .We show that and give almost matching lower and upper bounds for by showing that with suitable constants , for all sufficiently large . 相似文献
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In this paper, we study a multiple-terminal extension of the classic Hamiltonian path problem where salesmen are initially located at different depots and finally stopped at different terminals. To the best of our knowledge, only 2-approximation algorithm is available in the literature. For arbitrary , we first present a Christofides-like heuristic with a tight approximation ratio of . Besides, we also develop a -approximation algorithm by divide-and-conquer technique. The -approximation algorithm runs in polynomial time for fixed and . 相似文献
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Let be a simple graph, and let , and denote the maximum degree, the average degree and the chromatic index of , respectively. We called edge--critical if and for every proper subgraph of . Vizing in 1968 conjectured that if is an edge--critical graph of order , then . We prove that for any edge--critical graph , that is, This result improves the best known bound obtained by Woodall in 2007 for . 相似文献
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《Discrete Mathematics》2020,343(10):112010
Let be the -partite multigraph in which each part has size , where two vertices in the same part or different parts are joined by exactly edges or edges, respectively. It is proved that there exists a maximal set of edge-disjoint Hamilton cycles in for , the upper bound being best possible. The results proved make use of the method of amalgamations. 相似文献