Even-cycle decompositions of graphs with no odd-K4K_4-minor

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio

The Biosynthesis of Artemisinin (Qinghaosu) and the Phytochemistry of Artemisia annua L. (Qinghao)

The Chinese medicinal plant Artemisia annua L. (Qinghao) is the only known source of the sesquiterpene artemisinin (Qinghaosu), which is used in the treatment of malaria. Artemisinin is a highly oxygenated sesquiterpene, containing a unique 1,2,4-trioxane ring structure, which is responsible for the antimalarial activity of this natural product. The phytochemistry of A. annua is dominated by both sesquiterpenoids and flavonoids, as is the case for many other plants in the Asteraceae family. However, A. annua is distinguished from the other members of the family both by the very large number of natural products which have been characterised to date (almost six hundred in total, including around fifty amorphane and cadinane sesquiterpenes), and by the highly oxygenated nature of many of the terpenoidal secondary metabolites. In addition, this species also contains an unusually large number of terpene allylic hydroperoxides and endoperoxides. This observation forms the basis of a proposal that the biogenesis of many of the highly oxygenated terpene metabolites from A. annua - including artemisinin itself may proceed by spontaneous oxidation reactions of terpene precursors, which involve these highly reactive allyllic hydroperoxides as intermediates. Although several studies of the biosynthesis of artemisinin have been reported in the literature from the 1980s and early 1990s, the collective results from these studies were rather confusing because they implied that an unfeasibly large number of different sesquiterpenes could all function as direct precursors to artemisinin (and some of the experiments also appeared to contradict one another). As a result, the complete biosynthetic pathway to artemisinin could not be stated conclusively at the time. Fortunately, studies which have been published in the last decade are now providing a clearer picture of the biosynthetic pathways in A. annua. By synthesising some of the sesquiterpene natural products which have been proposed as biogenetic precursors to artemisinin in such a way that they incorporate a stable isotopic label, and then feeding these precursors to intact A. annua plants, it has now been possible to demonstrate that dihydroartemisinic acid is a late-stage precursor to artemisinin and that the closely related secondary metabolite, artemisinic acid, is not (this approach differs from all the previous studies, which used radio-isotopically labelled precursors that were fed to a plant homogenate or a cell-free preparation). Quite remarkably, feeding experiments with labeled dihydroartemisinic acid and artemisinic acid have resulted in incorporation of label into roughly half of all the amorphane and cadinane sesquiterpenes which were already known from phytochemical studies of A. annua. These findings strongly support the hypothesis that many of the highly oxygenated sesquiterpenoids from this species arise by oxidation reactions involving allylic hydroperoxides, which seem to be such a defining feature of the chemistry of A. annua. In the particular case of artemisinin, these in vivo results are also supported by in vitro studies, demonstrating explicitly that the biosynthesis of artemisinin proceeds via the tertiary allylic hydroperoxide, which is derived from oxidation of dihydroartemisinic acid. There is some evidence that the autoxidation of dihydroartemisinic acid to this tertiary allylic hydroperoxide is a non-enzymatic process within the plant, requiring only the presence of light; and, furthermore, that the series of spontaneous rearrangement reactions which then convert thi allylic hydroperoxide to the 1,2,4-trioxane ring of artemisinin are also non-enzymatic in nature

Chemical Composition and Larvicidal Activity of the EssentialOil of Laurus nobilisL. from Iran: Essential oil of Laurus nobilis

The chemical composition of the essential oil obtained from the aerial parts of Laurus nobilisL. has been examined by gas chromatography (GC) and GC- mass spectrometery (MS). The main components of the oil were identified. 1,8-Cineole was the major component in the oil together with ß-terpinyl acetate, terpinene-4-ol, α-pinene, ß-pinene, p-cymene, linalool and terpinene-4-ylacetate. The essential oil was tested against Anopheles stephensi and Culex pipienslarvae. The results obtained show that the essential oil could be considered as natural larvicidal agents

Triangulations of the torus with at most two odd vertices: structure and coloring

This thesis consists of two parts. In the first part, we give a simple geometric description of the set $mathcal G(5,5,8)$ of toroidal triangulations, all of whose vertices have degree six, except for two of degree five and one of degree eight. The motivation for studying such family is provided by Gr"unbaum coloring application described below. Each such triangulation is described by a cut-and-glue construction starting from an infinite triangular grid. In particular, we show that the members of $mathcal G(5,5,8)$ are obtained from a toroidal 6-regular graph (three parameters) by cutting out a special disk, described with two parameters, and ``stitching" along the cut. To achieve that, we develop some techniques and define some invariants to study the cycles of toroidal triangulations. We also introduce some special triangulated disks called ``blocks" and show how to detect their presence in a triangulation. Then, we show the existence of a special path that identifies the ``cut". Also, the graphs in $mathcal G(5,5,8)$ are classified into several families, based on the existence of some special cycles, containing a vertex of degree eight. Each family is, further, described by a schema for gluing together a few blocks. The second part regards coloring. A Gr"unbaum coloring of a graph $G$ which triangulates a surface is a 3-edge-coloring of $G$ in which every face is incident to three edges of different colors. In 1968 Gr"unbaum conjectured a generalization of the Four Color Theorem: every simple triangulation of every orientable surface has a Gr"unbaum coloring. In 2008 Kochol discovered counterexamples to Gr"unbaum\u27s conjecture on every orientable surface of genus at least five. Gr"unbaum\u27s conjecture is still believed to be true for the torus. We verify ``weak" Gr"unbaum conjecture for three families of triangulations in higher surfaces that, to our knowledge, are the only known families of triangulations with unbounded facewidth that are not 4-colorable. Also, as an application of our description, we propose a method by which to verify the Gr"unbaum\u27s conjecture for $mathcal G(5,5,8)$

Strongly Even-Cycle Decomposable Graphs

A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.SCOPUS: ar.jFLWINinfo:eu-repo/semantics/publishe

Silver Cubes

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n − 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K d n,I,c) where I is a maximum independent set in a Cartesian power of the complete graph Kn, and c: V (K d n) → {1,2,...,d(n − 1) + 1} is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory