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We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any . 相似文献
4.
Given a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7]. 相似文献
5.
《Discrete Mathematics》2022,345(12):113069
The toughness of a noncomplete graph G is the maximum real number t such that the ratio of to the number of components of is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant such that every graph with toughness at least is hamiltonian, is still open for general graphs. A graph is called -free if it does not contain any induced subgraph isomorphic to , the disjoint union of and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for -free graphs by showing that every 7-tough -free graph on at least three vertices is hamiltonian. 相似文献
6.
In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer , we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for , we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles. 相似文献
7.
受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[X_n]=ε[X_n]=0放宽为|E[X_n]|+|ε[X_n]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理. 相似文献
8.
A milestone in probability theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables with mean 0 and variance 1 In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph H. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random k‐uniform hypergraphs, we obtain the Central Limit Theorem and LIL for the number of Hamilton cycles. 相似文献
9.
《Discrete Mathematics》2022,345(2):112676
The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set and edge set . We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if , 2, 10, 20, 28, or . 相似文献
10.
By 5-h reaction of cis-[RuIICl2(DMSO)4] (M2) with K10[α2-P2W17O61] (M3) in ice-cooled, HCl-acidic aqueous solution, a water-soluble 1:2-type diamagnetic ruthenium(II) complex of formula K18[RuII(DMSO)2(P2W17O61)2]·35H2O (M1) was unexpectedly obtained as an analytically pure, homogeneous tan-colored solid, in which two DMSO ligands are coordinated to the ruthenium(II) atom. The cytotoxic potential of the complex was tested on C33A, DLD-1, and HepG-2 cancer cells and human normal embryonic lung fibroblasts cell MRC-5; the viability of the treated cells was evaluated by MTT assay. The mode of cell death was assessed by morphological study of DNA damage and apoptosis assays. Compound M1 induced cell death in a dose-dependent manner, and the mode of cell death was essentially apoptosis though necrosis was also noticed. Cell cycle analysis by flow cytometry indicated that M1 caused cell cycle arrest and accumulated cells in S phase. 相似文献