Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted . A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we show that . Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α^′(G). We also establish that , and show that for every tree T.     

[2+2]-Photocycloaddition von Cyclohexen an 2-Acetyl-5,5-dimethyl-1,3-cyclohexandion und 3-Acetyl-1,5,5-trimethyl-2,4-pyrrolidindion

Summary Irradiation of solutions of excess cyclohexene and 2-acetyl-5,5-dimethyl-1,3-cyclohexanedione (1), and 3-acetyl-1,5,5-trimethyl-2,4-pyrrolidinedione (4) results mainly in the formation of 1,5-diones2 and5. These originate from intermediate cycloadducts of cyclohexene and theexo-enols of the cyclic 1,3-diketones. The yields decrease with increasing polarity of the solvent. In solution2 and5 are in equilibrium with the cyclic hemiacetales3 and6.

     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Tetracyanoborate salts M[b(CN)4] with M = singly charged cations: properties and structures

A series of tetracyanoborate salts M[B(CN)4] with the singly charged cations of Li+, Na+, Rb+, Cs+, [NH4]+, Tl+, and Cu+ as well as the THF solvate tetracyanoborates Na[B(CN)4] x THF and [NH4][B(CN)4] x THF were synthesized and their X-ray structures, vibrational spectra, solubilities in water, and thermal stabilities determined and compared with already known M[B(CN)4] salts. Crystallographic data for these compounds are as follows: Na[B(CN)4], cubic, Fd3m, a = 11.680(1) A, Z= 8; Li[B(CN)4], cubic, P43m, a = 5.4815(1) A, Z= 1; Cu[B(CN)4], cubic, P43m, a = 5.4314(7) A, Z= 1; Rb[B(CN)4], tetragonal, /4(1)/a, a = 7.1354(2) A, c= 14.8197(6) A, Z= 4; Cs[B(CN)4], tetragonal, /4(1)/a, a = 7.300(2) A, c = 15.340(5) A, Z= 4; [NH4][B(CN)4], tetragonal, /4(1)/a, a = 7.132(1) A, c = 14.745(4) A, Z= 4; Tl[B(CN)4], tetragonal, /4(1)/a, a = 7.0655(2) A, c = 14.6791(4) A, Z= 4; Na[B(CN)4] x THF, orthorhombic, Pnma, a = 13.908(3) A, b = 9.288(1) A, c = 8.738(1) A, Z= 4; [NH4][B(CN)4] x THF, orthorhombic, Pnma, a = 8.831(1) A, b = 9.366(2) A, c = 15.061(3) A, Z= 4. The cubic Li+, Na+, and Cu+ salts crystallize in a structure consisting of two interpenetrating independent tetrahedral networks of M cations and [B(CN)4]- ions. The compounds with the larger countercations (Rb+, Cs+, Tl+, and [NH4]+) crystallize as tetragonal, also with a network arrangement. The sodium and ammonium salts with the cocrystallized THF molecules are both orthorhombic but are not isostructural. In the vibrational spectra the two CN stretching modes A1 and T2 coincide in general and the band positions are a measure for the strength of the interionic interaction. An interesting feature in the Raman spectrum of the copper salt is the first appearance of two CN stretching modes.     

Kombinierte Gas-Chromatographie-KMR-Technik

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The 7‐bromo derivative of 2‐amino‐2′‐deoxytubercidin fluorinated at the sugar moiety

The title compound, 2,4‐diamino‐5‐bromo‐7‐(2‐deoxy‐2‐fluoro‐β‐d ‐arabinofuranosyl)‐7H‐pyrrolo[2,3‐d]pyrimidine, C₁₁H₁₃BrFN₅O₃, shows two conformations of the exocyclic C4′—C5′ bond, with the torsion angle γ (O5′—C5′—C4′—C3′) being 170.1 (3)° for conformer 1 (occupancy 0.69) and 60.7 (7)° for conformer 2 (occupancy 0.31). The N‐glycosylic bond exhibits an anti conformation, with χ = −114.8 (4)°. The sugar pucker is N‐type (C3′‐endo; ³T₄), with P = 23.3 (4)° and τ_m = 36.5 (2)°. The compound forms a three‐dimensional network that is stabilized by several intermolecular hydrogen bonds (N—H...O, O—H...N and N—H...Br).     

Photochemical free radical formation from aliphatic dicarboxylic acids in solution as studied by electron spin resonance spectroscopy

Various free radicals formed during UV.-irradiation of aliphatic dicarboxylic acids in aqueous and methanolic solution are identified by ESR.-spectroscopy. Their structures point to α-cleavage and photoreduction as the dominant primary photochemical decay processes. The relative contributions of these reactions to the overall photodecomposition depend on solvent and degree of α-alkylation of the acid. Emission ESR.-spectra are found for radicals formed by C, CO-bond cleavage of α-dimethyl substituted acids. The polarization is referred to the triplet mechanism of CIDEP. and indicates this cleavage reaction occurs from a triplet molecular state.     

Zur spektrophotometrischen Bestimmung von ppm-Mengen Nickel in hochgereinigtem Wolfram und Wolframtrioxid benutzt

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