Publikationen

Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider “local” versions of such statements. In 1966, for instance, D. W. Barnette proved that a 3-connected planar graph contains a spanning tree of maximum degree at most 3. A local translation of this statement is that if G is a planar graph, X is a subset of specified vertices of G such that X cannot be separated in G by removing two or fewer vertices of G, then G has a tree of maximum degree at most 3 containing all vertices of X. Our results constitute a general machinery for strengthening statements about k-connected graphs (for 1 ≤ k ≤ 4) to locally spanning versions, that is, subgraphs containing a set X ⊆ V (G) of a (not necessarily planar) graph G in which only X has high connectedness. Given a graph G and X ⊆ V (G), we say M is a minor of G rooted at X, if M is a minor of G such that each bag of M contains at most one vertex of X and X is a subset of the union of all bags. We show that G has a highly connected minor rooted at X if X ⊆ V (G) cannot be separated in G by removing a few vertices of G. Combining these investigations and the theory of Tutte paths in the planar case yields locally spanning versions of six well-known results about degree-bounded trees, Hamiltonian paths and cycles, and 2-connected subgraphs of graphs.

Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H = Kr+1(r) with r ≥ 3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ (nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n − ⌊log2(n−2)⌋ − 6. However, if H is K5(3) minus an edge, then the process can run for Θ (n3) steps.

Optical frequency combs (OFCs) have great potential in communications, especially in dense wavelength-division multiplexing. However, the size of traditional OFCs based on conventional optical microcavities or dispersion fibers is at least tens of micrometers, far larger than that of nanoscale electronic chips. Therefore, reducing the size of OFCs to match electronic chips is of necessity. Here, for the first time to our knowledge, we introduce surface plasmon polaritons (SPPs) to the construction of OFCs to realize a miniature device. The thickness of our device is reduced below 1 μm. Though the presence of SPPs may induce ohmic and scattering loss, the threshold of the device is obtained as 9 μW, comparable to the conventional device. Interestingly, the response time is 13.2 ps, much faster than the optical counterparts. This work provides a feasible strategy for the miniaturization of OFCs.

Recent progress in nanotechnology has attracted interest to a biomedical application of the carbon nanoparticle C60 fullerene (C60) due to its unique structure and versatile biological activity. The dual functionality of C60 as a photosensitizer and a drug nanocarrier sets an opportunity to improve the efficiency of chemotherapeutic drugs for cancer cells. Pristine C60 demonstrates time-dependent accumulation with predominant mitochondrial localization in cancer cells. Nanomolar amounts of C60-drug nanocomplexes in 1:1 and 2:1 molar ratios improve the efficiency of cell treatment, complementing it with photodynamic approach. The cooperative enhancement interactions between mechanisms of chemo- and photodynamic therapies contribute to the obtained synergistic effect (namely “1+1>2”). A strong synergy of treatments arising from the combination of C60-mediated drug delivery and C60 photoexcitation indicates that a combination of chemo- and photodynamic treatments with C60-drug nanoformulations could provide a promising synergetic approach for cancer treatment.

Here, we present a protocol for the one-step synthesis of the title compound in quantitative yield using adapted Vilsmeier conditions. The product was characterized by 1H-,2H-,13C-NMR-, as well as IR and Raman spectroscopy. Spectral data are given in detail.