A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups

We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Achievable sets, brambles, and sparse treewidth obstructions

One consequence of the graph minor theorem is that for every k there exists a finite obstruction set Obs(TW?k). However, relatively little is known about these sets, and very few general obstructions are known. The ones that are known are the cliques, and graphs which are formed by removing a few edges from a clique. This paper gives several general constructions of minimal forbidden minors which are sparse in the sense that the ratio of the treewidth to the number of vertices n does not approach 1 as n approaches infinity. We accomplish this by a novel combination of using brambles to provide lower bounds and achievable sets to demonstrate upper bounds. Additionally, we determine the exact treewidth of other basic graph constructions which are not minimal forbidden minors.     

Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

Three New Diarylpropanes from Dioscorea composita

Three new diarylpropanes were isolated from the dried rhizomes of Dioscorea composita Hemsl., and their structures were determined as 1, 3-bis-(2-hydroxy-4-methoxyphenyl) propane (1), 1, 3-bis-(2, 4-dihydroxyphenyl)propane (2), 1-(2′-hydroxy-4′-O-β-D- glucopymnosy-phenyl)-3-(2“, 4“-dihydroxyphenyl)propane (3), by spectroscopic analysis, respectively.     

三七水溶性化学成分及其药理研究新进展

综述了近年来三七水溶性化学成分及其药理作用特别是对神经生理活性研究的进展，对三七与人参展其它植物植物进行了比较，阐明了三七所含多肽哆肽成分研究的意义及可行性。     

毛蕊杜鹃挥发油化学成分研究

毛蕊杜鹃(Rhododendron tubulosum Ching ex wang wei-yei)系杜鹃花科杜鹃花属植物,分布于四川、西藏、甘肃和青海,其中青海分布最广,藏民常作为单方草药用于治疗肺炎和老年慢性气管炎,具有镇咳、祛痰作用,本文用自制的同时水汽蒸馏-溶剂萃取装置,从采自青海玉树州的毛蕊杜鹃鲜叶和嫩枝提取挥发油,用毛细管气相色谱-质谱和Kovats指数双重定性法对     

Absolute configurations and circular dichroism of sesquiterpene-coumarin ethers

The absolute stereochemistry of 14 naturally occurring sesquiterpenecoumarin ethers is discussed. In 13 cases the coumarin moiety is isofraxidin (7-hydroxy-6,8-dimethoxycoumarin), in one case scopoletin (7-hydroxy-6-methoxycoumarin). The sesquiterpene moieties belong either to the bicyclic drimenole series or to its consecutive products (methyl migration, ring opening). The CD spectra of all compounds are given. The assignments of the absolute configurations are based on the method ofHoreau, chemical correlation by degradation reactions, extensive comparison of CD data, and biogenetic considerations.

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Absolute Konfigurationen und Circulardichroismus von Sesquiterpen-Cumarin-Ethern

Zusammenfassung Es wird die absolute Stereochemie von 14 natürlich vorkommenden Sesquiterpen-Cumarin-Ethern diskutiert. In 13 Fällen wird die Cumarineinheit von Isofraxidin (7-Hydroxy-6,8-dimethoxycumarin) repräsentiert, in einem Fall von Scopoletin (7-Hydroxy-6-methoxycumarin). Der Sesquiterpenteil ist entweder von Drimenol oder Folgeprodukten (Methylgruppenwanderung, Ringöffnung) abgeleitet. Die CD-Spektren aller Verbindungen sind angegeben. Die Zuordnung der absoluten Konfigurationen basiert auf der Methode vonHoreau, chemischer Korrelation über Abbaureaktionen, Vergleich der CD-Spektren und biogenetischen Überlegungen.

     

The influence of minor components on the structural reorganization in glassforming materials

Correlation of structure parameters of glasses and related crystals formed in homogeneous or heterogeneous nucleation processes by thermal treatment is discussed on the basis of DTA, TG and EGA measurements in relation to the textural patterns of the materials. For cordierite glass, crystallization of metastable disordered cordierite polymorphs is related to an exothermic heat evolution and simultaneous with a small weight loss (appr. 0.025%). By MS-EGA, evolution of water was determined during the transformation of the metastable melt to a metastable intermediate crystalline phase. Interpretation of the crystallization by comparing the available structure parameters of cordierite glasses and crystals alone is insufficient to explain the role of water in the kinetics of crystallization. Optical and electron microscopy of the primary crystallization phenomena show the metastable solid solution with low quartz-type structure. Interpretation of the crystallization behaviour in terms of conventional theory of nucleation and crystal growth is impossible.