NMR spectroscopy of organolithium compounds,part XXXIII: trans‐2,3‐dimethylcyclopropyllithium aggregation in diethylether: An equilibrium between three different complexes of comparable energy and the influence of LiBr on aggregate structure

The aggregate formation of trans‐2,3‐dimethylcyclopropyllithium ( 2 ) was studied in diethylether (DEE). With the help of the isotopic fingerprint method three clusters, a monomer, a dimer, and a fluxional tetramer in the ratio 1.00:0.53:0.22 were identified at 187 K. In the presence of 1 equivalent of LiBr cyclopropyllithium ( 1 ) forms in DEE at 163 K a mixed dimer and a mixed tetramer, while in THF a mixed dimer dominates. Under the same conditions 2 and 2,2,3,3‐tetramethylcyclopropyllithium ( 3 ) form in the solvent mixture DEE/THF (1:1) mixed dimers, as does 2 also in THF.     

Making an H $H$-free graph k $k$-colorable

We study the following question: How few edges can we delete from any $H$-free graph on $n$ vertices to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For $H$ any fixed odd cycle, we determine the answer up to a constant factor when $n$ is sufficiently large. We also prove an upper bound when $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.