Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V(G) = P, that the outer cycle of G coincides with the convex hull Conv(P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv(P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv(P) is even and that any two consecutive points on Conv(P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv(P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of \({\frac{n}{2}}\) red and \({\frac{n}{2}}\) blue points in general position, then adding Steiner points Q with \({|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}\) , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove \({|Q| \leq \frac{7n+34m-48}{18}}\) , where |P| = n and the number of points of P in Conv(P) is 2m.     

Ring size selective synthesis of meso-aryl expanded porphyrins

meso-Aryl expanded porphyrins were prepared in a ring size selective manner from methanesulfonic acid-catalyzed reaction of dipyrromethane and tripyrromethane with aryl aldehydes.     

Discrete dihedral-angle modulation in porphyrin wheels adsorbed on Cu(1 0 0) observed by scanning tunneling microscopy

Conformations of two dodecameric porphyrin wheels adsorbed on a Cu(1 0 0) were probed by using scanning tunneling microscopy (STM). Whereas a wheel consisting of six meso-meso linked diporphyrins was detected as uniform ring structure, several different images with three discrete molecular heights were detected for a wheel consisting of six meso-meso, β-β,β-β triply-linked planar diporphyrins. These results indicate that the former has a conformation similar to that in a free space, while the latter has various conformations with respect to orientation of planar diporphyrin units toward the metal surface. Several discrete STM images of the latter have been interpreted in terms of possible eight conformations, which vary as to relative orientation of neighboring diporphyrin units.     

Fused Corrole Dimers Interconvert between Nonaromatic and Aromatic States through Two‐Electron Redox Reactions

A singly linked corrole dimer was synthesized by condensation of a dipyrromethane‐1‐carbinol with 1,1,2,2‐tetrapyrroethane. Oxidation of the dimer gave doubly linked corrole dimers 9 and 10 as the first examples of fused corrole dimers involving a meso–meso linkage. Dimers 9 and 10 exhibit characteristic ¹H NMR spectra, absorption spectra, excited‐state dynamics, and two‐photon absorption (TPA) values, which indicate the nonaromatic nature of 9 and the aromatic nature of 10 . Interestingly, 9 is fairly stable despite its unusual 2H‐corrole structure, which has been ascribed to the presence of two direct connections between the individual corrole units.