Progress on perfect graphs

 A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle. Received: December 19, 2002 / Accepted: April 29, 2003 Published online: May 28, 2003 Key words. Berge graph – perfect graph – skew partition Mathematics Subject Classification (1991): 05C17     

Fluorinated nitrogen heterocycles via cyclization. III. 3-trifluoromethyl-1-(4-trifluoromethyl-2-pyridyl)-pyrazoles from fluorinated 1,3-diketones and 4-trifluoromethyl-2-hydrazinopyridines

Fluorinated 1,3-diketone compounds having the general structure CF₃COCH₂COR₁ underwent cyclization with 4-trifluoromethyl-2-hydrazinopyridines (IV) to give a series of 3-trifluoro-methyl-1-(4-trifluoromethyl-2-pyridyl)-pyrazoles (V). The hydrazinopyridine intermediates (IV) were prepared by conversion of 4-trifluoromethyl-2-pyridones (II) to the chloropyridines (III) followed by treatment of the latter with hydrazine hydrate.     

Graph minors. I. Excluding a forest

The path-width of a graph is the minimum value ofk such that the graph can be obtained from a sequence of graphsG₁,…,G_r each of which has at mostk + 1 vertices, by identifying some vertices ofG_i pairwise with some ofG_i+1 (1 ≤ i < r). For every forestH it is proved that there is a numberk such that every graph with no minor isomorphic toH has path-width?k. This, together with results of other papers, yields a “good” algorithm to test for the presence of any fixed forest as a minor, and implies that ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite.     

A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

Slowed picosecond kinetics of hot photogenerated carriers in GaAs

The hot photoluminescent kinetics of GaAs under intense picosecond excitations (10²⁸ photons/cm² sec) have been measured. A slow risetime of the near bandedge luminescence has been observed arising from a slowed cooling of the electron distribution. The slowed electron kinetics of over 50 fold are attributed to the screening of the electron-phonon interaction. Carrier densities at these excitation intensities are limited by saturation of the absorption. These results are important for understanding and may help clarify the mechanisms of laser annealing.