Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.

d-Regular Graphs of Acyclic Chromatic Index at Least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) <= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+2 colors. We also show that a'(K-n,K-n) >= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d >= 5, n >= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Conformations of dehydrophenylalanine containing peptides: nmr studies of an acyclic hexapeptide with two delta Z-Phe residues

The conformation of an acyclic dehydrophenylalanine (delta Z-Phe) containing hexapeptide, Boc-Phe-delta Z-Phe-Val-Phe-delta Z-Phe-Val-OMe, has been investigated in CDCl3 and (CD3)2SO by 270-MHz 1H-nmr. Studies of NH group solvent accessibility and observation of interresidue nuclear Overhauser effects (NOEs) suggest a significant solvent-dependent conformational variability. In CDCl3, a population of folded helical conformations is supported by the inaccessibility to solvent of the NH groups of residues 3-6 and the detection of several NiH----Ni + 1H NOEs. Evidence is also obtained for conformational heterogeneity from the detection of some Ci alpha H----Ni + 1H NOEs characteristic of extended strands. In (CD3)2SO, the peptide largely favors an extended conformation, characterized by five solvent-exposed NH groups and successive Ci alpha H----Ni + 1H NOEs for the L-residues and Ci beta H----Ni + 1H NOEs for the delta Z-Phe residues. The results suggest that delta Z-Phe residues do not provide compelling conformational constraints.

Metabolism of a monoterpene ketone, R-(+)-pulegone—a hepatotoxin in rat

R-(+)-Pulegone was administered orally to rats and the urinary metabolites were investigated. Six metabolites were isolated and purified using column and thin layer chromatographic techniques. Metabolites were identified by i.r., n.m.r. and mass spectral analyses.The neutral metabolites isolated from urine of rats treated with pulegone (I) were: pulegol (II), 2-hydroxy-2(1'-hydroxy-1'-methylethyl)-5-methylcyclohexanone (III), 3,6-dimethyl-7a-hydroxy-5,6,7,7a-tetrahydro-2(4H)-benzofuranone (V) and menthofuran (VII). Metabolites II and III were also excreted in conjugated form.Acidic metabolites isolated from urine of rats treated with pulegone (I) were: 5-methyl-2(1'-methyl-1'-carboxyethylidene)cyclohexanone (IV) and 5-methyl-5-hydroxy-2(1'hydroxy-'-carboxyethyl)cyclohexanone (VI).

Conformations of dehydrophenylalanyl containing peptides. Nuclear Overhauser effect study of two acyclic

Two isomeric, acyclic tetrapeptides containing a Z-dehydrophenylalanine residue (Δz-Phe) at position 2 or 3, Boc-Leu-Ala-Δz-Phe-Leu-OMe (1) and Boc-Leu-Δz-Phe-Ala-Leu-OMe (2), have been synthesized and their solution conformations investigated by 270MHz 1H n.m.r. spectroscopy. In peptide 1 the Leu(4) NH group appears to be partially shielded from solvent, while in peptide 2 both Ala(3) and Leu(4) NH groups show limited solvent accessibility. Extensive difference nuclear Overhauser effect (n.O.e.) studies establish the occurrence of several diagnostic inter-residue n.O.e.s (CαjH ⇆ Ni+1H and NiH ⇆ Ni+1H) between backbone protons. The simultaneous observation of “mutually exclusive” n.O.e.s suggests the presence of multiple solution conformations for both peptides. In peptide 1 the n.O.e. data are consistent with a dynamic equilibrium between an -Ala-Δz-Phe- Type II β-turn structure and a second species with Δz-Phe adopting a partially extended conformation with Ψ values of ± 100° to ± 150°. In peptide 2 the results are compatible with an equilibrium between a highly folded consecutive β-turn structure for the -Leu-Δz-Phe-Ala- segment and an almost completely extended conformation.

Biochemical, histopathological and ultrastructural changes in rat liver induced by r-( + )-pulegone, a monoterpene ketone

Biochemical, histopathological and ultrastructural changes occurring at different time points after intraperitoneal administration of a single dose of pulegone (300 mg/kg) were studied. Significant decreases in the level of liver microsomal cytochrome P-450 (67%), heme (37%), aminopyrine N-demethylase (60%) and glucose-6-phosphatase (58%), were noticed 24 hr after pulegone treatment. Alanine amino transferase (ALT) levels increased in a time dependent manner, following exposure of rats to pulegone. Light microscopic studies of liver tissues showed dilation of central veins and distention of sinusoidal spaces 6 hr after pulegone treatment. Initial centrilobular necrosis was noticed at 12 hr. Centrilobular necrosis became severe at 18 hr and nuclear changes included karyorrhexis and karyolysis. Midzonal and periportal degenerative changes in addition to centrilobular necrosis was observed 24 hr after pulegone administration. Electron microscopic changes showed severe degeneration of endoplasmic reticulum, swelling of mitochondria and nuclear changes, 24 hr after administration of pulegone. The time course profile of the hepatocytes after treatment with pulegone indicates that endoplasmic reticulum is the organelle most affected, following which other degenerative changes occur ultimately leading to cell death.

Destruction Of Rat-Liver Microsomal Cytochrome-P-450 Invitro By A Monoterpene Ketone, Pulegone A Hepatotoxin

Significant destruction (68%) of liver microsomal cytochrome P-450 and homogeneous cytochrome P-450 purified from PB-treated rats is noticed upon incubation with 10 mM pulegone at 37-degrees-C for 30 min. There is also a concomitant loss of heme. The destructive phenomenon does not require metabolic activation of pulegone. The destruction of purified cytochrome P-450 is time-dependent and saturable. Structure-activity studies suggest that an alpha-isopropylidine ketone unit with a methyl positioned para to the isopropylidine group as in pulegone is necessary for the in vitro destruction of cytochrome P-450. SKF-525A at a concentration of 4-mM obliterates the destruction of cytochrome P-450 by pulegone. Experiments with C-14-pulegone suggest that pulegone or its rearranged product binds covalently to the prosthetic heme of cytochrome P-450.

Studies on the metabolism of a monoterpene ketone, R-(+)-pulegone-a hepatotoxin in rat: isolation and characterization of new metabolites

1. The metabolic disposition of R-(+)-pulegone (1) was examined in rats following four daily oral doses (250 mg/kg). 2. Six metabolites, namely pulegol (II), 2-hydroxy-2-(1-hydroxy-1-methylethyl)-5-methylcyclohexanone (III), 3,6-dimethyl-7a-hydroxy-5,6,7,7a-tetrahydro-2(4H)-benzofuranone (IV), menthofuran (V), 5-methyl-2-(1-methyl-1-carboxyethylidene)cyclohexanone (VI), and 5-methyl-5-hydroxy-2-(1-hydroxy-1-carboxyethyl)cyclohexanone (VII) have previously been isolated from rat urine, and identified (Moorthy et al. (1989a). Eight new metabolites have now been isolated from rat urine, namely, 5-hydroxy-pulegone (VIII), piperitone (IX), piperitenone (X), 7-hydroxy-piperitone (XI), 8-hydroxy piperitone (XII), p-cresol (XIII), geranic acid (XIV) and neronic acid (XV). These were identified by n.m.r., i.r. and mass spectrometry. 3. Based on these results, metabolic pathways for the biotransformation of R-(+)-pulegone in rat have been proposed.

Effects of menthofuran, a monoterpene furan on rat liver microsomal enzymes, in vivo

Oral administration (250 mg/kg) of menthofuran, a monoterpene furan, to rats once daily for 3 days caused hepatotoxicity as judged by a significant increase in serum glutamate pyruvate transaminase (SGPT) and decreases in glucose-6-phosphatase and aminopyrine N-demethylase activities. Administration of menthofuran also resulted in a decrease in the levels of liver microsomal cytochrome P-450, whereas cytochrome b(5) and NAD(P)H-cytochrome c reductase activities were not affected. These effects of menthofuran were both dose- and time-dependent. Pretreatment of rats with phenobarbital (PB) prior to menthofuran treatment potentiated hepatotoxicity suggesting that a PB-induced cytochrome P-450 catalyzed the formation of reactive metabolite(s) responsible for the hepatotoxicity.

Utility of microbes in organic-synthesis - selective transformations of acyclic isoprenoids by aspergillus-niger

Bioconversion of acyclic isoprenoids using a strain of Aspergillus niger results in hydroxylated metabolites with regio- and stereoselectivity. The organism carries out oxidation of the terminal allylic methyl group and the remote double bond in all the compounds tested (I-VII). However, these two activities seem to have preferential structural requirements. When an acyclic isoprenoid with a ketone functionality such as geranylacetone is used as the substrate, the organism also carries out the asymmetric reduction of the keto group. All the metabolites formed have been purified and characterized by conventional spectroscopic methods and quantification has been made by gas chromatographic analyses.

Metabolic disposition of a monoterpene ketone, piperitenone, in rats: Evidence for the formation of a known toxin, p-cresol

It was shown earlier that the monoterpene ketone, piperitenone (I) is one of the major metabolites of R-(+)-pulegone, a potent hepatotoxin, In the present studies, the metabolic disposition of piperitenone (I) was examined in rats. Piperitenone (I) was administered orally (400 mg/kg of the b. wt./day) to rats for 5 days, The following urinary metabolites were isolated and identified by various spectral analyses: p-cresol (VI), 6,7-dehydromenthofuran (III), p-mentha-1,3,5,8-tetraen-3-ol (IX), p-mentha-1,3, 5-friene-3, 8-diol (X), 5-hydroxypiperitenone (VIII), 7-hydroxypiperitenone (XI), 10-hydroxypiperitenone (XII), and 4-hydroxypiperitenone (VII). Incubation of piperitenone (I) with phenobarbital-induced rat liver microsomes in the presence of NADPH resulted in the formation of five metabolites which have been tentatively identified as metabolites III, VII, VIII, XI, XII, on the basis of gas chromatography retention time and gas chromatography-mass spectrometry analysis. Based on these results, a probable mechanism for the formation of p-cresol from piperitenone (I) via the intermediacy of metabolite III has been proposed.

Acyclic Edge-Coloring of Planar Graphs

A proper edge-coloring with the property that every cycle contains edges of at least three distinct colors is called an acyclic edge-coloring. The acyclic chromatic index of a graph G, denoted. chi'(alpha)(G), is the minimum k such that G admits an acyclic edge-coloring with k colors. We conjecture that if G is planar and Delta(G) is large enough, then chi'(alpha) (G) = Delta (G). We settle this conjecture for planar graphs with girth at least 5. We also show that chi'(alpha) (G) <= Delta (G) + 12 for all planar G, which improves a previous result by Fiedorowicz, Haluszczak, and Narayan Inform. Process. Lett., 108 (2008), pp. 412-417].

On network coding for acyclic networks with delays

Problems related to network coding for acyclic, instantaneous networks (where the edges of the acyclic graph representing the network are assumed to have zero-delay) have been extensively dealt with in the recent past. The most prominent of these problems include (a) the existence of network codes that achieve maximum rate of transmission, (b) efficient network code constructions, and (c) field size issues. In practice, however, networks have transmission delays. In network coding theory, such networks with transmission delays are generally abstracted by assuming that their edges have integer delays. Using enough memory at the nodes of an acyclic network with integer delays can effectively simulate instantaneous behavior, which is probably why only acyclic instantaneous networks have been primarily focused on thus far. However, nulling the effect of the network delays are not always uniformly advantageous, as we will show in this work. Essentially, we elaborate on issues ((a), (b) and (c) above) related to network coding for acyclic networks with integer delays, and show that using the delay network as is (without adding memory) turns out to be advantageous, disadvantageous or immaterial, depending on the topology of the network and the problem considered i.e., (a), (b) or (c).