Degree Sums and Covering Cycles

It is shown that if in a simple graph G of order n the sum of degrees of any three pairwise non-adjacent vertices is at least n, then there are two cycles (or one cycle and an edge or a vertex) of GF that contain all the vertices. © 1995 John Wiley & Sons, Inc.     

Influence of Aliphatic Chain Length on Structural,Thermal and Electrochemical Properties of n-alkylene Benzyl Alcohols: A Study of the Odd–Even Effect

The century-old, well-known odd–even effect phenomenon is still a very attractive and intriguing topic in supramolecular and nano-scale organic chemistry. As a part of our continuous efforts in the study of supramolecular chemistry, we have prepared three novel aromatic alcohols (1,2-bis[2-(hydroxymethyl)phenoxy]butylene (Do4OH), 1,2-bis[2-(hydroxymethyl)phenoxy]pentylene (Do5OH) and 1,2-bis[2-(hydroxymethyl)phenoxy]hexylene (Do6OH)) and determined their crystal and molecular structures by single-crystal X-ray diffraction. In all compounds, two benzyl alcohol groups are linked by an aliphatic chain of different lengths (CH₂)_n; n = 4, 5 and 6. The major differences in the molecular structures were found in the overall planarity of the molecules and the conformation of the aliphatic chain. Molecules with an even number of CH₂ groups tend to be planar with an all-trans conformation of the aliphatic chain, while the odd-numbered molecule is non-planar, with partial gauche conformation. A direct consequence of these structural differences is visible in the melting points—odd-numbered compounds of a particular series display systematically lower melting points. Crystal and molecular structures were additionally studied by the theoretical calculations and the melting points were correlated with packing density and the number of CH₂ groups. The results have shown that the generally accepted rule, higher density = higher stability = higher melting point, could not be applied to these compounds. It was found that the denser packaging causes an increase in the percentage of repulsive H‧‧‧H interactions, thereby reducing the stability of the crystal, and consequently, the melting points. Another interesting consequence of different molecular structures is their electrochemical and antioxidative properties—a non-planar structure displays the highest oxidation peak of hydroxyl groups and moderate antioxidant activity.     

Contractions and minimalk-colorability

Coloring the vertex set of a graphG with positive integers, thechromatic sum (G) ofG is the minimum sum of colors in a proper coloring. Thestrength ofG is the largest integer that occurs in every coloring whose total is(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strengths has at least ((2 + ) ^s–1 – (2 – ) ^s–1)/ vertices (s 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for everys 3 there exist precisely two treesT _s andR _s such that every tree of strength at leasts is edge-contractible toT _s orR _s .     

Marking Games and the Oriented Game Chromatic Number of Partial <Emphasis Type="Italic">k</Emphasis>-Trees

 Nešetřil and Sopena introduced the concept of oriented game chromatic number. They asked whether the oriented game chromatic number of partial k-trees was bounded. Here we answer their question positively. Received: January 12, 2001 Final version received: February 25, 2002     

Adoption of the Numerical Method of Chebyshev Polynomials to the Stability Analysis of Delayed DEs

We propose a new way of the application of the numerical method based on Chebyshev approximation concerning the stability analysis of (periodic) delay‐differential equations. The technique is tested on the delayed Mathieu equation.     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

Intersection Dimensions of Graph Classes

The intersection dimension of a graphG with respect to a classA of graphs is the minimumk such thatG is the intersection of at mostk graphs on vertex setV(G) each of which belongs toA. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) ifA is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect toA is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.:On grid intersection graphs, Discrete Math.87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.     

The Marica-Schonheim Inequality in Lattices

The Marica-Schönheim Inequality says that if A is a finitefamily of sets, then |A–||A| where A–A=[A₁\A₂:A₁,A₂A]. For a finite lattice L and AL, we define a–b=(J_a\J_b)where J_a=[jL:ja and j is join-irreducible], and if AL then welet A–A=[a₁–a₂: a₁, a₂A]. Then the analogue of theMarica-Schöonheim Inequality is |A–A|A| for all AL.We prove that this is true if L is distributive or complementedand modular or L is a partition lattice.     

Observation of a diamagnetic signal up to 132 K in a Tl−Ca−Ba−Cu−O compound

Tl–Ca–Ba–Cu–O compounds of the nominal composition of (1, 1, 1, 1) have been prepared from different starting materials and with the same heat treatment process. During the investigation of their properties a diamagnetic signal has been found up to the temperature of 132 K. This hints at the existence of a certain material content distributed into discrete superconducting domains. The parameters of it reflect a superstructure of the (2, 2, 2, 3) and (2, 1, 2, 2) compositions having a time-dependent behaviour. On the basis of the signal levels at 132 K and 4.2 K, the quantity of the high temperature superconducting material is about 0.01% of the total one at liquid He temperatures.     

Towards a superstatistical SU(2) Yang-Mills EoS

We study an SU(2) lattice gauge field system on a 10⁴ symmetric and on a 10³ × 2 asymmetric lattice in order to learn about the equation of state.