Some results on the Oberwolfach problem

Aequationes mathematicae -     

Fluorescent Rhodamines and Fluorogenic Carbopyronines for Super‐Resolution STED Microscopy in Living Cells

A range of bright and photostable rhodamines and carbopyronines with absorption maxima in the range of λ=500–630 nm were prepared, and enabled the specific labeling of cytoskeletal filaments using HaloTag technology followed by staining with 1 μm solutions of the dye–ligand conjugates. The synthesis, photophysical parameters, fluorogenic behavior, and structure–property relationships of the new dyes are discussed. Light microscopy with stimulated emission depletion (STED) provided one‐ and two‐color images of living cells with an optical resolution of 40–60 nm.     

Far‐Red Emitting Fluorescent Dyes for Optical Nanoscopy: Fluorinated Silicon–Rhodamines (SiRF Dyes) and Phosphorylated Oxazines

Far‐red emitting fluorescent dyes for optical microscopy, stimulated emission depletion (STED), and ground‐state depletion (GSDIM) super‐resolution microscopy are presented. Fluorinated silicon–rhodamines (SiRF dyes) and phosphorylated oxazines have absorption and emission maxima at about λ≈660 and 680 nm, respectively, possess high photostability, and large fluorescence quantum yields in water. A high‐yielding synthetic path to introduce three aromatic fluorine atoms and unconventional conjugation/solubilization spacers into the scaffold of a silicon–rhodamine is described. The bathochromic shift in SiRF dyes is achieved without additional fused rings or double bonds. As a result, the molecular size and molecular mass stay quite small (<600 Da). The use of the λ=800 nm STED beam instead of the commonly used one at λ=750–775 nm provides excellent imaging performance and suppresses re‐excitation of SiRF and the oxazine dyes. The photophysical properties and immunofluorescence imaging performance of these new far‐red emitting dyes (photobleaching, optical resolution, and switch‐off behavior) are discussed in detail and compared with those of some well‐established fluorophores with similar spectral properties.     

List Homomorphisms and Circular Arc Graphs

G , H, and lists , a list homomorphism of G to H with respect to the lists L is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Received: July 22, 1996/Revised: Revised June 10, 1998     

Heterogeneous catalytic 1,4-addition of arylmagnesium compounds to chalcones

Copper(II) on a 4?Å molecular sieve support catalyses the chemoselective addition of alkyl- and arylmagnesium halides to chalcones. Only the 1,4-addition products were obtained in high yields.