Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

Porphyrin-Stabilized Nitrenium Dication

An azepine-fused Ni^II-porphyrin dimer was synthesized by oxidative amination of β–β linked Ni^II-porphyrin dimer, and its N-aryl congener was synthesized by twofold Buchwald–Hartwig amination of β-to-β linked and dichlorinated Ni^II-porphyrin dimer. Oxidation of the NH congener gave neutral aminyl radical and nitrenium ion successively in a similar manner to the previously reported NH-bridged Ni^II-porphyrin dimer. Oxidation of the N-aryl congener afforded formal nitrenium dication in a single-step two-electron oxidation. Stable nitrenium dications that are isoelectronic to neutral trivalent boron compounds and tertiary carbocations are rare. The chemical stability of the nitrenium dication was improved by chlorine substitution at the most reactive β-position. The structural, optical, magnetic, electrochemical properties of these compounds were fully characterized.     

β‐to‐β 2,5‐Pyrrolylene‐Linked Cyclic Porphyrin Oligomers

β‐to‐β 2,5‐Pyrrolylene linked cyclic porphyrin oligomers have been synthesized by Suzuki–Miyaura coupling of 2,5‐diborylpyrrole and 3,7‐dibromoporphyrin. The cyclic porphyrin oligomers exhibit roughly coplanar structures, strong excitonic coupling, small electrochemical HOMO–LUMO gaps, and ultrafast excitation energy transfer between the neighboring porphyrins via the pyrrolylene bridge.     

Synthesis of Peripherally Nitrated,Aminated, and Arylaminated Subporphyrins

2‐Nitro‐5,10,15‐tri(4‐tert‐butylphenyl)subporphyrin 2 was prepared by the nitration of 5,10,15‐tri(4‐tert‐ butylphenyl)subporphyrin 1a with five equivalents of Cu(NO₃)₂ ? 5 H₂O in a mixed EtOAc/Ac₂O solution and was reduced into 2‐amino‐5,10,15‐tri(4‐tert‐butylphenyl)subporphyrin 3 . Bromination of 5,10,15‐triphenylsubporphyrin 1b with 1.5 equivalents of N‐bromosuccinimide (NBS) gave 2‐bromo‐5,10,15‐triphenylsubporphyrin, which was converted into various 2‐arylamino‐5,10,15‐triphenylsubporphyrins ( 4a , 4b , 4c , 4d ) and 2‐benzamido‐5,10,15‐triphenylsubporphyrin 5 through Pd‐catalyzed cross‐coupling reactions. These molecules constitute the first examples of mono‐β‐substituted subporphyrins. These subporphyrins exhibit significantly perturbed optical and electrochemical properties, which reflect a large influence of the peripherally attached substituents on the electronic networks of subporphyrins.