Electroluminescent segmented liquid crystalline trimers

The synthesis and liquid crystal behaviour of light‐emitting trimeric liquid crystals consisting of three mesomorphic moieties connected by aliphatic spacers are reported. The combination of an aromatic light‐emitting central core and two cholestanyl (dihydrocholesteryl) groups induces a helical liquid crystalline phase with circularly polarised photoluminescence and electroluminescence. These segmented trimers are designed to possess a high glass transition temperature below which the structure of liquid crystalline phases can be fixed. The effects of odd and even spacers, spacer length and the presence of large lateral substituents on the liquid crystal behaviour and the glass transition temperature of these trimers were studied. Electroluminescence from a segmented liquid crystalline trimer is reported for the first time.     

2‐factors with prescribed and proscribed edges

Let G be a finite k‐edge‐connected simple graph. We consider when a set of independent edges can be extended to a 2‐factor such that this 2‐factor avoids a fixed set of independent edges. A complete characterization is provided in those cases, where this is feasible. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 48–58, 2005     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

N-extendability of symmetric graphs

It is proved that a cyclically (k ? 1)(2n ? 1)-edge-connected edge transitive k-regular graph with even order is n-extendable, where k ≥ 3 and k ? 1 ≥ n ≥ ?(k + 1)/2?. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc.