The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

Porphyrins with exocyclic rings. 14. Synthesis of tetraacenaphthoporphyrins, a new family of highly conjugated porphyrins with record-breaking long-wavelength electronic absorptions

The Soret band for porphyrins is usually observed in the near-ultraviolet at approximately 400 nm, and few examples of "nonexpanded" porphyrins with this major absorption band at values above 500 nm have previously been reported in the literature. Ring fusion with aromatic ring systems such as naphthalene, anthracene, or phenanthrene generally only produces minor bathochromic shifts to this diagnostic absorption band. In this paper, the synthesis of a series of tetraacenaphthoporphyrins and their metal chelates is reported. The compact nature of the acenaphthylene ring system allows the introduction of meso substituents using the Lindsey methodology. meso-Tetraphenylporphyrin 10a shows the presence of a Soret band at 556 nm, while p-methoxy and p-nitro substituents in 10f and 10g, respectively, further shift this band to 560 and 570 nm. Addition of TFA produces the corresponding dications with slightly higher wavelength Soret bands at 565, 573, and 588 nm. These values compare to 525 nm for the dication of tetraacenaphthylene 8, which lacks the meso-aryl substituents, indicating that steric crowding and its resulting distortion of the macrocyclic conformation is responsible for a significant albeit minor portion of these shifts. The nickel(II), copper(II), and zinc chelates of 10a produce Soret bands at 528, 545, and 558 nm, respectively, demonstrating that the trend for increasing red shifts in metalloporphyrins across the periodic table is retained for this series. The lead(II) chelate 19d gave an additional "hyper" shift that brought the Soret band to 604 nm. A similar red shift could be achieved by introducing four phenylethynyl substituents at the meso positions, and this highly conjugated porphyrin (20) also showed a Soret band at 604 nm, while the corresponding dication afforded this absorption band at 629 nm. The essentially additive "hyper" shift due to lead chelation brought the Soret band for the related lead(II) complex 22d to 642 nm. These effects are by far the largest ever observed for true porphyrins and demonstrate that the Soret band can be fined tuned to virtually any part of the visible spectrum.