New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Some results on the existence of large sets of t‐designs

A set of trivial necessary conditions for the existence of a large set of t‐designs, LS[N](t,k,ν), is for i = 0,…,t. There are two conjectures due to Hartman and Khosrovshahi which state that the trivial necessary conditions are sufficient in the cases N = 2 and 3, respectively. Ajoodani‐Namini has established the truth of Hartman's conjecture for t = 2. Apart from this celebrated result, we know the correctness of the conjectures for a few small values of k, when N = 2 and t ≤ 6, and also when N = 3 and t ≤ 4. In this article, we show that similar results can be obtained for infinitely many values of k. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 144–151, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10027     

Incomplete perfect mendelsohn designs with block size 4 and one hole of size 7

Let v,k, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k-IPMD(v,n), is a triple (X, Y, ??) where X is a v-set (of points), Y is an n-subset of X, and ?? is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X × X)?(Y × Y) appears t-apart in exactly one block of ?? and no ordered pair (a,b) ∈ Y × Y appears in any block of ?? for any t, where 1 ≤ t ≤ k ? 1. In this article, we obtain conclusive results regarding the existence of 4-IPMD(v,7) where the necessary conditions are v = 2 or 3(mod 4) and v ≥ 22. We also provide an application to the problem relating to coverings of PMDs with block size 4. © 1993 John Wiley & Sons, Inc.     

Coloring graphs from lists with bounded size of their union

A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 2^4k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Toric Rings Arising from Cyclic Polytopes

Let d and n be positive integers with n ≥ d + 1 and 𝒫 ? ? ^d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[?_≥0𝒜_𝒫] denote its associated semigroup K-algebra, where 𝒜_𝒫 = {(1, α) ∈ ? ^d+1: α ∈ 𝒫} ∩ ? ^d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R ₁), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only.     

定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Un Algorithme pour le Calcul des Syzygies sur V[X] dans le cas où V est un Domaine de Valuation

Soit V un domaine de valuation. Nous donnons un algorithme pour calculer une base du V-saturé d'un sous-module de type fini d'un V-module libre (avec une base éventuellement infinie). Nous l'appliquons pour calculer le V-saturé d'un sous-V[X]-module de type fini de V[X] ⁿ (n ∈ ?*). Ceci permet enfin de calculer un système générateur fini pour les syzygies sur V[X] d'une famille finie de vecteurs de V[X] ^k .

We give an algorithm for computing the V-saturation of any finitely generated submodule of V[X] ⁿ (n ∈ ?*), where V is a valuation domain. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of V[X] ^k .     

Incomplete perfect mendelsohn designs with block size 4 and holes of size 2 and 3

Let v, k, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k-IPMD(v, n), is a triple (X, Y, ??) where X is a v-set (of points), Y is an n-subset of X, and ?? is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X × X)\(Y × Y) appears t-apart in exactly one block of ?? and no ordered pair (a,b) ∈ Y × Y appears in any block of ?? for any t, where 1 ≤ t ≤ k ? 1. In this article, the necessary conditions for the existence of a 4-IPMD(v, n), namely (v ? n) (v ? 3n ? 1) ≡ 0 (mod 4) and v ≥ 3n + 1, are shown to be sufficient for the case n = 3. For the case n = 2, these conditions are sufficient except for v = 7 and with the possible exception of v = 14,15,18,19,22,23,26,27,30. The latter result provides a useful application to the problem relating to the packing of perfect Mendelsohn designs with block size 4. © 1994 John Wiley & Sons, Inc.     

Constructions for rotational near resolvable block designs

A (ν, k, k?1) near resolvable block design (NRBD) is r‐rotational over a group G if it admits G as an automorphism group of order (ν?1)/r fixing exactly one point and acting semiregularly on the others. We give direct and recursive constructions for rotational NRBDs with particular attention to 1‐rotational ones. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 157–181, 2001     

A small generating set for the 3‐BD closed set of block sizes ≥ 6

Let v, k be positive integers and k ≥ 3, then K_k = : {v: v ≥ k} is a 3‐BD closed set. Two finite generating sets of 3‐BD closed sets K₄ and K₅ are obtained by H. Hanani [5] and Qiurong Wu [12] respectively. In this article we show that if v ≥ 6, then v ∈ B₃(K,1), where K = {6,7,…,41,45,46,47,51,52,53,83,84}\{22,26}; that is, we show that K is a generating set for K₆. Finally we show that v ∈ B₃(6,20) for all v ∈ K\{35,39,40,45}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 128–136, 2008     

A Class of Multiplier Hopf Algebras

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism. Presented by A. Verschoren.     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Maximal sets of triangle-factors on v = 6m vertices

We show that the necessary condition m ≤ k ≤ 3m − 1 that there exists a maximal set of k triangle-factors on 6m ≥ 18 vertices is also sufficient, except possibly when k = m. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 235–244, 1998     

Taylor series and divisibility

LetA be an augmentedK-algebra; defineT:A →A ?k _kA byT(a)=1?a ?a?1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA?k _kA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

On the equations of the edge cone of a graph¶and some applications

Let G be a graph such that none of its components is bipartite. We describe the facets of the cone generated by the columns of the incidence matrix of G. Let k[G] be the subring generated by the monomials of degree two defining the edges of G, where k is a field. Some estimates for the a-invariant of k[G] are shown when G is the cone of a normal connected non bipartite graph or G is the join of two normal connected non bipartite graphs. Received: 24 July 1997 / Revised version: 3 March 1998     

Rings of Constants of the Form k[f]

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ? k[X]? k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f] in k[X]. A similar construction for fields of rational functions is also given.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001