Graph family operations

In previous papers, Catlin introduced four functions, denoted , , , and , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.     

Fitting Quantiles: Doubling,HR, HQ,and HHH Distributions

Abstract

This article introduces a family of distributional shapes which is flexible in the sense that it contains skewed and symmetric laws as well as heavy-tailed and light-tailed laws. The proposed family is also practically convenient because it is easy to fit to a table of quantiles from any distribution. Inversely, for each of the distributional shapes it is trivial to compute quantiles for any desired probability, and it is possible to compute the corresponding densities.     

Chain Intersecting Families

Let be a family of subsets of an n-element set. is called (p,q)-chain intersecting if it does not contain chains and with . The maximum size of these families is determined in this paper. Similarly to the p = q = 1 special case (intersecting families) this depends on the notion of r-complementing-chain-pair-free families, where r = p + q − 1. A family is called r-complementing-chain-pair-free if there is no chain of length r such that the complement of every set in also belongs to . The maximum size of such families is also determined here and optimal constructions are characterized. The first author is a member of the Egerváry Research Group (EGRES). Research is supported by OTKA grants T 037547 and TS 049788, by European MCRTN Adonet, Contract Grant No. 504438 and by the Egerváry Research Group of the Hungarian Academy of Sciences. The work of the second author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant numbers T037846 and NK62321.     

Constructions and bounds for separating hash families

In this paper, we present a new construction for strong separating hash families by using hypergraphs and obtain some optimal separating hash families. We also improve some previously known bounds of separating hash families.     

Differential inequalities, normality and quasi-normality

We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality.     

A classification of the structures of some Sperner families and superimposed codes

In this paper we will give a new proof by using group action to prove the uniqueness of maximal Sperner families of [n]. We will also prove the uniqueness of Sperner families F of [n] with by using a combinatorial approach. Furthermore, by using the uniqueness of Sperner family, we will classify all the structures of (1,2) superimposed codes of size 9×10 and 9×11.     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

Admissible values of one parameter for maximal Sperner families of subsets of the type (<Emphasis Type="Italic">k,k</Emphasis> + 1)

In this paper we generalize one assertion (obtained by us earlier) on admissible values of a certain parameter for partial maximal Sperner families (m. S. f.) of subsets of a finite set of the type (k, k + 1). We also prove that the minimal value of the parameter under consideration for all m. S. f. of the type (k, k + 1), except for two families, is less than \(\left( {_k^{n - 1} } \right)\) ? 1.     

A Katona-type proof of an Erd?s-Ko-Rado-type theorem

Let p?1/2 and let μ_p be the product measure on {0,1}ⁿ, where μ_p(x)=p^∑x_i(1-p)^n-∑x_i. Let A⊂{0,1}ⁿ be an intersecting family, i.e. for every x,y∈A there exists 1?i?n such that x_i=y_i=1. Then μ_p(A)?p. Our proof uses a probabilistic trick first applied by Katona to prove the Erd?s-Ko-Rado theorem.     

贪婪t－相交有限序列族

设Ｆ为有限序列族，对ａ＝（ａ１，ａ２，…，ａｎ）∈Ｆ，ａｉ为整数且０≤ａｉ≤ｓｉ（整数），记ｓ（ａ）＝｛ｊ｜１≤ｊ≤ｎ，ａｊ＞０｝，ｓ（Ｆ）＝｛ｓ（ａ）｜ａ∈Ｆ｝，及Ａ｛１，２，…，ｎ｝时Ｗ（Ａ）＝Пｉ∈Ａｓｉ．称Ｆ为贪婪ｔ－相交，如对任何ａ，ｂ∈Ｆ，至少有ｔ个ａｉ，ｂｉ＞０，且Ｗ（Ａ）≥Ｗ（（｛１，２，…，ｎ｝－Ａ）＋Ｂ）对任何Ａ∈Ｓ（Ｆ）及ＢＡ（｜Ｂ｜＝ｔ－１）成立．本文得到当ｓ１＞ｓ２＞…＞ｓｎ时的最大贪婪ｔ－相交有限序列族．     

Internal and external partial difference families and cyclotomy

We introduce the concept of a disjoint partial difference family (DPDF) and an external partial difference family (EPDF), a natural generalization of the much-studied disjoint difference family (DDF), external difference family (EDF) and partial difference set (PDS). We establish properties and indicate connections to other recently-studied combinatorial structures. We show how DPDFs and EPDFs may be formed from PDSs, and present various cyclotomic constructions for DPDFs and EPDFs. As part of this, we develop a unified cyclotomic framework, which yields some known results on PDSs, DDFs and EDFs as special cases.     

Admissibility of prediction intervals

Admissibility of prediction intervals is considered in a specified family. It is shown that the best invariant prediction interval is strongly admissible in a location family and in a scale family. Though the similar result has not been obtained for a location and scale family, the best invariant prediction interval for a normal distribution is shown to be weakly admissible.     

The spectrum of (<Emphasis Type="Italic">gv</Emphasis>, <Emphasis Type="Italic">g</Emphasis>, 3, λ)-DF in <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">gv</Emphasis></Subscript>

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)-difference family in Z _gv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type g ^v without short orbits is determined. This work was supported by National Natural Science Foundation of China (Grants Nos. 10771013, 10831002)     

Almost Connected Orders

One of the standard axioms for semiorders states that no three-point chain is incomparable to a fourth point. We refer to asymmetric relations satisfying this axiom as almost connected orders or ac-orders. It turns out that any relation lying between two weak orders, one of which covers the other for inclusion, is an ac-order (albeit of a special kind). Every ac-order is bracketed in a natural way by two weak orders, one the maximum in the set of weak orders included in the ac-order, and the other minimal, but not necessarily the minimum, in the set of weak orders that include the ac-order. The family of ac-orders on a finite set with at least five elements is not well graded (in the sense of Doignon and Falmagne, 1997). However, such a family is both upgradable and downgradable, as every nonempty ac-order contains a pair whose deletion defines an ac-order on the same set, and for every ac-order which is not a chain, there is a pair whose addition gives an ac-order.     

The spectrum of (gv, g, 3, λ)-DF in Z_(gv)

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)- difference family in Zgv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type gv without short orbits is determined.