Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009     

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     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an i‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph K_v to be i‐perfect if for every edge [x, y] of K_v there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ‐decomposition (Γ‐factorization) of K_v that is i‐perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge‐colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i‐perfect Γ‐decomposition of K_v provided that Γ is an i‐equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009     

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non‐negative integer. We say G is k‐antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted‐k‐antimagic if for any vertex weight function w: V→?, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well‐known conjecture asserts that every connected graph G≠K₂ is 0‐antimagic. On the other hand, there are connected graphs G≠K₂ which are not weighted‐1‐antimagic. It is unknown whether every connected graph G≠K₂ is weighted‐2‐antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted‐2‐antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted‐1‐antimagic. We also prove that every connected graph G≠K₂ on n vertices is weighted‐ ?3n/2?‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

On fractional K‐factors of random graphs

Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non‐negative real‐valued function f : 𝒦→ ℝ₊ is called a fractional K‐factor if ∑ _{K:v∈K∈𝒦}f(K) ≤ 1 for every v ∈ V and ∑ _K∈𝒦f(K) = n/r. For a non‐empty graph K let d(K) = e(K)/v(K) and d⁽¹⁾(K) = e(K)/(v(K) ‐ 1). We say that K is strictly K₁‐balanced if for every proper subgraph K^′ ⊊ K, d⁽¹⁾(K^′) < d⁽¹⁾(K). We say that K is imbalanced if it has a subgraph K^′ such that d(K^′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K₁‐balanced, then with probability tending to 1 as n →∞, at the first moment τ₀ when every vertex is covered by a copy of K, the graph has a fractional K‐factor. This result is the best possible. As a consequence, if K is K₁‐balanced, we derive the threshold probability function for a random graph to have a fractional K‐factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ₀ and the appearance of a fractional K‐factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

In this paper, we look at the existence of (v K) pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}. For K = {5, 7}, {5, 8}, {5, 7, 9}, we reduce the largest v for which a (v, K)‐PBD is unknown to 639, 812, and 179, respectively. When K is Q_≥8, the set of all prime powers ≥ 8, we find several new designs for 1,180 ≤ v ≤ 1,270, and reduce the largest unsolved case to 1,802. For K =Q_0,1,5(8), the set of prime powers ≥ 8 and ≡ 0, 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of {8, 9} or Q_0,1(8), the set of prime powers ≡ 0 or 1 (mod 8). © 2004 Wiley Periodicals, Inc.     

Transformation graph

The transformation graph G^-+- of a graph G is the graph with vertex set V(G)E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: (i) u,vV(G) and they are not adjacent in G, (ii) u,vE(G) and they are adjacent in G, (iii) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. In this paper, for any graph G, we determine the connectivity and the independence number of G^-+-. Furthermore, for a graph G of order n4, we show that G^-+- is hamiltonian if and only if G is not isomorphic to any graph in {2K₁+K₂,K₁+K₃}{K_1,n-1,K_1,n-1+e,K_1,n-2+K₁}.     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resolvable even‐cycle systems with a 1‐rotational automorphism

In this article, necessary and sufficient conditions for the existence of a 1‐rotationally resolvable even‐cycle system of λK_v are given, which are eventually for the existence of a resolvable even‐cycle system of λK_v. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 394–407, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10058     

A complete solution to the existence problem for 1‐rotational k‐cycle systems of Kv

The necessary and sufficient conditions for the existence of a 1‐rotational k‐cycle system of the complete graph K_v are established. The proof provides an algorithm able to determine, directly and explicitly, an odd k‐cycle system of K_v whenever such a system exists. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 283–293, 2009     

<Emphasis Type="Italic">v</Emphasis>-Adic maximal extensions,spectral norms and absolute Galois groups

Let (K, v) be a perfect rank one valued field and let ([`(K<sub>v</sub>)],[`(v)]){(\overline{K_{v}},\overline{v})} be the canonical valued field obtained from (K, v) by the unique extension of the valuation [(v)\tilde]{\widetilde{v}} of K _v , the completion of K relative to v, to a fixed algebraic closure [`(K<sub>v</sub>)]{\overline{K_{v}}} of K <sub>v</sub> . Let [`(K)]{\overline{K}} be the algebraic closure of K in [`(K<sub>v</sub>)]{\overline {K_{v}}}. An algebraic extension L of K, L ì [`(K)]{L\subset\overline{K}}, is said to be a v-adic maximal extension, if [`(v)] | <sub>L</sub>{\overline{v}\mid_{L}} is the only extension of v to L and L is maximal with this property. In this paper we describe some basic properties of such extensions and we consider them in connection with the v-adic spectral norm on [`(K)]{\overline{K}} and with the absolute Galois groups Gal([`(K)]/K){(\overline{K}/K)} and Gal([`(K_v)] /K_v){(\overline{K_{v}} /K_{v})}. Some other auxiliary results are given, which may be useful for other purposes.     

Subdivisions of K5 in Graphs Embedded on Surfaces With Face‐Width at Least 5

We prove that if G is a 5‐connected graph embedded on a surface Σ (other than the sphere) with face‐width at least 5, then G contains a subdivision of K₅. This is a special case of a conjecture of P. Seymour, that every 5‐connected nonplanar graph contains a subdivision of K₅. Moreover, we prove that if G is 6‐connected and embedded with face‐width at least 5, then for every v ∈ V(G), G contains a subdivision of K₅ whose branch vertices are v and four neighbors of v.     

Extremal results for rooted minor problems

In this article, we consider the following problem. Given four distinct vertices v₁,v₂,v₃,v₄. How many edges guarantee the existence of seven connected disjoint subgraphs X_i for i = 1,…, 7 such that X_j contains v_j for j = 1, 2, 3, 4 and for j = 1, 2, 3, 4, X_j has a neighbor to each X_k with k = 5, 6, 7. This is the so called “rooted K_{3, 4}‐minor problem.” There are only few known results on rooted minor problems, for example, [15,6]. In this article, we prove that a 4‐connected graph with n vertices and at least 5n ? 14 edges has a rooted K_3,4‐minor. In the proof we use a lemma on graphs with 9 vertices, proved by computer search. We also consider the similar problems concerning rooted K_3,3‐minor problem, and rooted K_3,2‐minor problem. More precisely, the first theorem says that if G is 3‐connected and e(G) ≥ 4|G| ? 9 then G has a rooted K_3,3‐minor, and the second theorem says that if G is 2‐connected and e(G) ≥ 13/5|G| ? 17/5 then G has a rooted K_3,2‐minor. In the second case, the extremal function for the number of edges is best possible. These results are then used in the proof of our forthcoming articles 7 , 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 191–207, 2007     

Cyclic k‐cycle systems of order 2kn + k: A solution of the last open cases

We exhibit cyclic (K_v, C_k)‐designs with v > k, v ≡ k (mod 2k), for k an odd prime power but not a prime, and for k = 15. Such values were the only ones not to be analyzed yet, under the hypothesis v ≡ k (mod 2k). Our construction avails of Rosa sequences and approximates the Hamiltonian case (v = k), which is known to admit no cyclic design with the same values of k. As a particular consequence, we settle the existence question for cyclic (K_v, C_k)‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004.     

Alternating groups and flag‐transitive 2‐(v,k, 4) symmetric designs

In this article, we study the classification of flag‐transitive, point‐primitive 2‐ (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag‐transitive, point‐primitive nontrivial 2‐ (v, k, 4) symmetric design ?? is an alternating group A_n for n≥5, then (v, k) = (15, 8) and ?? is one of the following: (i) The points of ?? are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A₇ or A₈, and the stabilizer G_x of a point x of ?? is L₃(2) or AGL₃(2), respectively. (ii) The points of ?? are the edges of the complete graph K₆ and the blocks are the complete bipartite subgraphs K_{2, 4} of K₆, G = A₆ or S₆, and G_x = S₄ or S₄ × Z₂, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011     

On the 3BD‐closed set B3({4,5,6})

Let B₃(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B₃(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B₃ ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B₃{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B₃({4,5,6}) by Hanani and that B₃({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B₃({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B₃({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.     

Minimum embedding of P3‐designs into (K4—e)‐designs

A (K₄ ? e)‐design on v + w points embeds a P₃‐design on v points if there is a subset of v points on which the K₄ ? e blocks induce the blocks of a P₃‐design. It is shown that w ≥ ¾(v ? 1). When equality holds, the embedding design is easily constructed. In this paper, the next case, when w = ¾v, is settled with finitely many exceptions. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 352–366, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10044     

Decompositions of the 3-uniform hypergraphs <Emphasis Type="Italic">K</Emphasis><Stack><Subscript>v</Subscript><Superscript>(3)</Superscript></Stack> into hypergraphs of a certain type

In this paper, we investigate a generalization of graph decomposition, called hypergraph decomposition. We show that a decomposition of a 3-uniform hypergraph K(3)v into a special kind of hypergraph K(3)4 - e exists if and only if v ≡ 0, 1, 2 (mod 9) and v ≥ 9.     

A MICRO‐LEVEL ‘CONSUMER APPROACH’ TO SPECIES POPULATION DYNAMICS

ABSTRACT. In this paper we develop a micro ecosystem model whose basic entities are representative organisms which behave as if maximizing their net offspring under constraints. Net offspring is increasing in prey biomass intake, declining in the loss of own biomass to predators and Allee's law applies. The organism's constraint reflects its perception of how scarce its own biomass and the biomass of its prey is. In the short‐run periods prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth functions. We are able to explicitly determine these growth functions for a simple food web when specific parametric net offspring functions are chosen in the micro‐level ecosystem model. For the case of a single species our model is shown to yield the well‐known Verhulst‐Pearl logistic growth function. With two species in predator‐prey relationship, we derive differential equations whose dynamics are completely characterized and turn out to be similar to the predator‐prey model with Michaelis‐Menten type functional response. With two species competing for a single resource we find that coexistence is a knife‐edge feature confirming Tschirhart's [2002] result in a different but related model.