Coupled choosability of plane graphs

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying for every , there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008     

Two‐factors each component of which contains a specified vertex

It is shown that if G is a graph of order n with minimum degree δ(G), then for any set of k specified vertices {v₁,v₂,…,v_k} ? V(G), there is a 2‐factor of G with precisely k cycles {C₁,C₂,…,C_k} such that v_i ∈ V(C_i) for (1 ≤ i ≤ k) if or 3k + 1 ≤ n ≤ 4k, or 4k ≤ n ≤ 6k ? 3,δ(G) ≥ 3k ? 1 or n ≥ 6k ? 3, . Examples are described that indicate this result is sharp. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 188–198, 2003     

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     

An improvement on H design

An H(m, g, 4, 3) is a triple , where X is a set of mg points, is a partition of X into m disjoint sets of size g, and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. Such a design was introduced by Hanani 2 , who used it to study Steiner quadruple systems. Mills showed that for , an H(m, g, 4, 3) exists if and only if mg is even and is divisible by 3, and that for , an H(5, g, 4, 3) exists if g is divisible by 4 or 6 10 . In this article, we shall show that an H(5, g, 4, 3) exists if g is even, and . © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 25–35, 2009     

Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The lotto numbers L(n,3,p,2)

An (n,k,p,t)‐lotto design is an n‐set N and a set of k‐subsets of N (called blocks) such that for each p‐subset P of N, there is a block for which . The lotto number L(n,k,p,t) is the smallest number of blocks in an (n,k,p,t)‐lotto design. The numbers C(n,k,t) = L(n,k,t,t) are called covering numbers. It is easy to show that, for n ≥ k(p ? 1), For k = 3, we prove that equality holds if one of the following holds:

• (i) n is large, in particular
• (ii)
• (iii) 2 ≤ p ≤ 6.

© 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006     

Arc‐chromatic number of digraphs in which every vertex has bounded outdegree or bounded indegree

A k‐digraph is a digraph in which every vertex has outdegree at most k. A ‐digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ (k) (resp. ) of the arc‐chromatic number over the k‐digraphs (resp. ‐digraphs). El‐Sahili [3] showed that . After giving a simple proof of this result, we show some better bounds. We show and where θ is the function defined by . We then study in more detail properties of Φ and . Finally, we give the exact values of and for . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 315–332, 2006     

Upper bounds on the general covering number Cλ(v,k, t,m)

A collection of k‐subsets (called blocks) of a v‐set X (v) = {1, 2,…, v} (with elements called points) is called a t‐(v, k, m, λ) covering if for every m‐subset M of X (v) there is a subcollection of with such that every block K ∈ has at least t points in common with M. It is required that v ≥ k ≥ t and v ≥ m ≥ t. The minimum number of blocks in a t‐(v, k, m, λ) covering is denoted by C_λ(v, k, t, m). We present some constructions producing the best known upper bounds on C_λ(v, k, t, m) for k = 6, a parameter of interest to lottery players. © 2004 Wiley Periodicals, Inc.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators ofthe form : ?²(|z |²) → L²(|z |²). Here, (f):= (Id–P_l )( ^kf) and Pl is the projection onto A_l ²(?, |z |²):= cl(span{ ^m zⁿ | m, n ∈ N, m ≤ l }). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators are not bounded for l < k – 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L² symbols, generalizing arguments from [6], seems possible and is planned for future work (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an online random k‐SAT model

Given n Boolean variables x₁,…,x_n, a k‐clause is a disjunction of k literals, where a literal is a variable or its negation. Suppose random k‐clauses are generated one at a time and an online algorithm accepts or rejects each clause as it is generated. Our goal is to accept as many randomly generated k‐clauses as possible with the condition that it must be possible to satisfy every clause that is accepted. When cn random k‐clauses on n variables are given, a natural online algorithm known as Online‐Lazy accepts an expected (1 ? )cn + a_kn clauses for some constant a_k. If these clauses are given offline, it is possible to do much better, (1 ? )cn + Ω( )n can be accepted whp . The question of closing the gap between a_k and Ω( ) for the online version remained open. This article shows that for any k ≥ 1, any online algorithm will accept less than (1 ? )cn + (ln 2)n k‐clauses whp , closing the gap between the constant and Ω( ). Furthermore we show that this bound is asymptotically tight as k → ∞. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

A randomized on–line algorithm for the k–server problem on a line

The k–server problem is one of the most important and well‐studied problems in the area of on–line computation. Its importance stems from the fact that it models many practical problems like multi‐level memory paging encountered in operating systems, weighted caching used in the management of web caches, head motion planning of multi‐headed disks, and robot motion planning. In this paper, we investigate its randomized version for which Θ(log k)–competitiveness is conjectured and yet hardly any <k competitive algorithms are known, even for the simplest of metric spaces of O(k) size. We present a –competitive randomized k–server algorithm against an oblivious adversary when the underlying metric space is given by n equally spaced points on a line . This algorithm is <k competitive for . Thus, it provides a super–linear bound for n with o(k)–competitiveness for the first time and improves the best results known so far for the range on . © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Nodes of large degree in random trees and forests

We study the asymptotic behavior of the number N_k,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that N_k,n is asymptotically normal if and asymptotically Poisson distributed if . If , then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Random partitions with restricted part sizes

For a subset of positive integers let Ω(n, ) be the set of partitions of n into summands that are elements of . For every λ ∈ Ω(n, ), let M _n(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on Ω(n, ), and regard M _n as a random variable. In this paper the limiting density of the (suitably normalized) random variable M _n is determined for sets that are sufficiently regular. In particular, our results cover the case = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some remarks about factors of graphs

A (g, f)‐factor of a graph is a subset F of E such that for all , . Lovasz gave a necessary and sufficient condition for the existence of a (g, f)‐factor. We extend, to the case of edge‐weighted graphs, a result of Kano and Saito who showed that if for any , then a (g, f)‐factor always exist. In addition, we use results of Anstee to provide new necessary and sufficient conditions for the existence of a (g, f)‐factor. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 265–274, 2008     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .