Lower bounds and upper bounds for chromatic polynomials

In this paper we give lower bounds and upper bounds for chromatic polynomials of simple undirected graphs on n vertices having m edges and girth exceeding g © 1993 John Wiley & Sons, Inc.     

Inclusion-exclusion: Which terms cancel?

We continue our study of the phenomenon of cancellation appearing in the inclusion-exclusion principle. More precisely, we regard the terms occuring in the inclusion-exclusion principle as nodes of a rooted tree and identify each set of cancelling terms with a subtree of this tree.     

Pd-Catalyzed Asymmetric N-Allylation of Amino Acid Esters with Exceptional Levels of Catalyst Control: Stereo-Divergent Synthesis of ProM-15 and Related Bicyclic Dipeptide Mimetics

A general and powerful method for the stereo-controlled Pd-catalyzed N-allylation of amino acid esters is reported, as a previously largely unsolved synthetic challenge. Employing a new class of tartaric acid-derived C₂-symmetric chiral diphosphane ligands the developed asymmetric amination protocol allows the conversion of various amino acid esters to the N-allylated products with highest levels of enantio- or diastereoselectivity in a fully catalyst-controlled fashion and predictable configuration. Remarkably, the in situ generated catalysts also exhibit outstanding levels of activity (ligand acceleration). The usefulness of the method was demonstrated in the stereo-divergent synthesis of a set of new conformationally defined dipeptide mimetics, which represent new modular building blocks for the development of peptide-inspired bioactive compounds.     

Bonferroni-type inequalities and binomially bounded functions

We present a unified approach to an important subclass of Bonferroni-type inequalities by considering the so-called binomially bounded functions. Our main result associates with each binomially bounded function a Bonferroni-type inequality. By appropriately choosing this function, several well-known and new results are deduced in a concise and unified way.     

An Upper Bound to the Chromatic Number of a Uniform Hypergraph without Cycles of Length Two or Three

In this paper, we establish a new upper bound to the chromatic number of a uniform hypergraph having no cycles of length two or three. The proof is based on a recent lower bound to the chromatic polynomial.