The forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The forcing spectrum of G is the collection of forcing numbers of all perfect matchings of G. In this paper, we classify the perfect matchings of a generalized Petersen graph P (n, 2) in two types, and show that the forcing spectrum is the un...

Let n > 2 be a positive integer and let φ denote Euler’s totient function. Define φ(n) = φ(n) and φ(n) = φ(φ(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = φ(n) + φ(n) + · · ·+ φ(n) + 1, where φ(n) = 2. We say n is a perfect totient number if S(n) = n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect ...

In this paper we introduce the notion of “strong n-perfect rings” which is in some way a generalization of the notion of “n-perfect rings”. We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a class of n-perfect rings that are not strong n-perfect rings. Finally, we establish the transfer of this notion to the direct product notions.

let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.

This article examines the geographical corridors for the establishment of rice in seventeenth-century Dutch Guiana. One corridor of introduction is associated with the expulsion of Dutch planters from Brazil in 1644, whose slaves reestablished longstanding subsistence preferences with their exodus to the colony. Another corridor links its introduction to the African Gold Coast, where rice devel...

An (N ;n,m, {w1, . . . , wt})-separating hash family is a set H of N functions h : X −→ Y with |X| = n, |Y | = m, t ≥ 2 having the following property. For any pairwise disjoint subsets C1, . . . , Ct ⊆ X with |Ci| = wi, i = 1, . . . , t, there exists at least one function h ∈ H such that h(C1), h(C2), . . . , h(Ct) are pairwise disjoint. Separating hash families generalize many known combinator...

We refer the reader to Jerrum's book [1] for the analysis of a Markov chain for generating a random matching of an arbitrary graph. Here we'll look at how to extend the argument to sample perfect matchings in dense graphs and arbitrary bipartite graphs. Both of these results are due to Jerrum and Sinclair [2]. Dense graphs We'll need some definitions and notations first: Definition 7.1 A graph ...

This paper investigates the construction of 8-QAM+ perfect arrays with at least an odd integer in their sizes, and gives the sufficient condition producing distinct 8-QAM+ perfect arrays. For different choices of odd integers in a given size of at least an odd integer, the resultant distinct 8-QAM+perfect arrays have their own sizes. For each choice, the number of the resultant distinct 8-QAM+p...