The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G. The main result is that χ_c(μ(G)) = χ(μ(G)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ_c(G) = and χ(G) = , consequently, there exist graphs G such that χ_c(G) is as close to χ(G) − 1 as you want, but χ_c(μ(G)) = χ(μ(G)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999     

Distinguishing two graph-encoded manifolds of Lins

Lins has conjectured that the two 3-manifolds that he refers to as H and H̄ are not homeomorphic. He suggests that their fundamental groups may be the same, but that they may be distinguishable by their quantum invariants. This article describes the proof that they, in fact, have different fundamental groups. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 298–302, 1999     

Graph minors and linkages

Bollobás and Thomason showed that every 22k‐connected graph is k‐linked. Their result used a dense graph minor. In this paper, we investigate the ties between small graph minors and linkages. In particular, we show that a 6‐connected graph with a K minor is 3‐linked. Further, we show that a 7‐connected graph with a K minor is (2,5)‐linked. Finally, we show that a graph of order n and size at least 7n?29 contains a K minor. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 75–91, 2005     

On the minimum size of tight hypergraphs

A k-graph, H = (V, E), is tight if for every surjective mapping f: V → {1,….k} there exists an edge α ? E sicj tjat f|_α is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number ? of edges in a tight k-graph with n vertices are given. We conjecture that ? for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow. © 1992 John Wiley & Sons, Inc.     

Star factorizations of graph products

A k‐star is the graph K_1,k. We prove a general theorem about k‐star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k‐star factorizations of any power (K_q)^s of a complete graph with prime power order q, products C × C ×··· × C of k cycles of arbitrary lengths, and any power (C_r)^s of a cycle of arbitrary length. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 59–66, 2001     

Enumeration of graphs by degree sequence

The degree sequence (d₀, d₁, …, d_p-1) of a graph G of order p is defined by d_k = the number of points of G of degree k. Methods of Robinson are extended to produce a generating function F(x₀, x₁, x₂, …) where the coefficient of x…x is the number of graphs of order p having degree sequence (d₀, …, d_p-1).     

On a conjecture on maximal planar sequences

Let d1 d2 dp denote the nonincreasing sequence d₁, …, d₁, d₂, …, d₂, …, d_p, …, d_p, where the term d_i appears k_i times (i = 1, 2, …, p). In this work the author proves that the maximal 2-sequences: 7³6¹5¹⁵, 7⁵6¹5¹⁷, 7⁷6¹5¹⁹ are planar graphical, in contrast to a conjecture by Schmeichel and Hakimi.     

The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□?□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q_n)=?log₂n?+1 which matches the lower bound, and that Det(K)=?log₃(2n+1)?+1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hⁿ)=Θ(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory     

On the Lorentz-Marcinkiewicz Operator Ideal

This article deals with the LORENTZ-MARCINKIEWICZ operator ideal ?? generated by an additive s-function and the LORENTZ-MARCINKIEWICZ sequence space λ^q(φ). We give eigenvalue distributions for operators belonging to ?? (E, E) and we show the interpolation properties of ??-ideals. Furthermore, we study certain SCHAUDER bases in ?? (H, K), H and K Hilbert spaces.     

Function Spaces with Exponential Weights II

This paper is a continuation of [8]. We study weighted function spaces of type B and F on the Euclidean space Rⁿ, where u is a weight function of at most exponential growth. In particular, u(χ (±|χ|) is an admissible weight. We deal with atomic decompositions of these spaces. Furthermore, we prove that the spaces B and F are isomorphic to the corresponding unweighted spaces B and F.     

On invariant measures of extensions of Markov transition functions

By using the LITTLEWOOD matrices A_2n we generalize CLARKSON' S inequalities, or equivalently, we determine the norms ‖A_2n: l(L_P) → l(L_P)‖ completely. The result is compared with the norms ‖A_2n: l → l‖, which are calculated implicitly in PIETSCH [6].     

Über die Radon-Transformation kreissymmetrischer Funktionen und ihre Beziehung zur Sommerfeldschen Theorie der Hankelfunktionen

For the Radon transform of functions with circular symmetry an inversion formula is proved in a new and elementary way. The inversion formula combined with Fourier theory is applied to Sommer-feld's integral for H, yielding a representation of products which generalizes Nicholson's integral for |H| ².     

ITERATIONS OF SATISFACTION CLASSES AND MODELS OF PEANO ARITHMETIC

We consider iterations of satisfaction classes and apply them to construct expansions of models of Peano arithmetic to models of A|Δ+∑-AC. 1991 MSC: 03F35, 03C62.     

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

We consider a domain Ω in ?ⁿ of the form Ω = ?^l × Ω′ with bounded Ω′ ? ?^n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)^m + (? ? ?… ? ?)^m]u = fe^?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^α (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.     

The Analytic Polynomial-Time Hierarchy

Motivated by results on interactive proof systems we investigate an ?-?hierarchy over P using word quantifiers as well as two types of set quantifiers. This hierarchy, which extends the (arithmetic) polynomial-time hierarchy, is called the analytic polynomial-time hierarchy. It is shown that every class of this hierarchy coincides with one of the following Classes: ∑, Π (k?0), PSPACE, ∑ or Π (k?1). This improves previous results by Orponen [6] and allows interesting comparisons with the above mentioned results on inter-active proof systems.     

An extremal problem for subdivisions of K

It is proved that every graph G with ‖G‖ ≥ 2|G| − 5, |G| ≥ 6, and girth at least 5, except the Petersen graph, contains a subdivision of K, the complete graph on five vertices minus one edge. © 1999 John Wiley & Sons, Inc, J. Graph Theory 30: 261–276, 1999     

Sharp Sobolev Embeddings and Related Hardy Inequalities: The Sub -- Critical Case

The paper deals with sharp embeddings of the spaces B and F into rearrangement-variant spaces and related Hardy inequalities. Here (1/_p, s) belongs to the interior of the shaded invariant spaces region in the Figure     

Selbstadjungierte Relations-Erweiterungen für eine Klasse von symmetrischen Differential-Funktional-Operatoren

Symmetry- and selfadjointness-conditions are derived for ordinary differential-integral-interface operators under integral-interface conditions. Criteria for the existence of selfadjoint extensions in L×L are given. These extensions are characterized in a constructive way. The main tools are some extension-theorems for linear relations (subspaces), wich are developed in section 2.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.