Sum List Coloring Graphs

Let G=(V,E) be a graph with n vertices and e edges. The sum choice number of G is the smallest integer p such that there exist list sizes (f(v):v ∈ V) whose sum is p for which G has a proper coloring no matter which color lists of size f(v) are assigned to the vertices v. The sum choice number is bounded above by n+e. If the sum choice number of G equals n+e, then G is sum choice greedy. Complete graphs K_n are sum choice greedy as are trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocks is sum choice greedy is also sum choice greedy. We also determine the sum choice number of K_2,n, and we show that every tree on n vertices can be obtained from K_n by consecutively deleting single edges where all intermediate graphs are sc-greedy.     

Muon spin relaxation in different samples of DyAl2

Two samples of DyAl₂ were studied by longitudial μSR in the paramagnetic and ferromagnetic regime. In a region T_c±40 K the samples give different signals although X-ray analysis can not detect any impurity phase in either of them.     

Zur intramolekularen nucleophilen Addition von 2,3-Dihydrodipyrrin-1(10H)-onen

Model compound5 was prepared by acid catalyzedintramolecular nucleophilic addition of the alcohol (Z)-4. In comparison to theintermolecular addition productl-2, differences in structure and reactivity could be determined by X-ray crystal structure analysis and¹H-NMR-spectroscopic investigations. The interpretation of these differences are based on the assumption of stereoelectronic advantages in the intermolecular adductl-2. Its methoxy group was found to beperiplanar with respect to the -(NCO)-plane of the lactam unit enabeling efficient delocalization of the amide lone pair to the *-orbital of the C-OMe-bond. The orientation of the corresponding C-O-bond in5 is fixed by the rigid structure of the bicyclic ring system and can be classified assynclinal. Consequentlyl-2 eliminates methanol very easily under acidic or high temperature conditions, whereas5 is a stable compound.In the stereoelectronically favoured adductl-2 n *-delocalization increases elimination reactivity on the one hand and thermodynamic stability on the other hand. Therefore adducts of this kind may be good candidates for biological regulators—a matter of interest with regard to the protein-chromophore interaction in biliproteins.

     

Hyperbolicity of the complement of plane curves

In this paper it is proved that the complement in ₂ of a holomorphic curve D of genus g2 is a hermitian hyperbolic complex manifold provided that certain conditions on the singularities of the dual D* of D are satisfied and that every tangent at D* intersects D* in at least two distinct points.Dedicated to Karl Stein