A connected graph G is called t-tough if t . w(G -S ) 6 JSJ for any subset S of V(G) with w(G -S ) > 1, where w(G -S ) is the number of connected components of G -S. We prove that every k-tough graph has a k-factor if klG( is even and IGI 2 k + 1. This result, first conjectured by Chvdtal, is sharp in the following sense: For any positive integer k and for any positive real number E , there exists a (k -&tough graph G with klGl even and (GI 2 k + 1 which has no k-factor.The toughness of a graph is an invariant first introduced by Chvatal. In [3] he mainly studied relations between toughness and the existence of hamiltonian cycles or k-factors, and stated several conjectures. In this article, we give answers to some of them.We consider only finite undirected graphs without loops and multiple edges. Let G be a graph. We denote by V ( G ) and E(G) the set of vertices and the set of edges, respectively. Let S and T be disjoint subsets of

Highly luminescent water-soluble InP/ZnS core-shell nanocrystals were prepared using a newly developed method that incorporates reactive phase transfer and photochemical processing. Poor-emitting InP nanocrystals (NCs, 2-4 nm) prepared solvothermally using toluene were transferred into alkaline aqueous solution containing thiol and zinc ions. When these NCs in aqueous solution were subsequently irradiated by ultraviolet (UV) light, they showed intense size-dependent photoluminescence (PL) from green to red due to the formation of a thick (more than 1 nm) ZnS shell on the NCs. The surface dissolution of the NCs, under conditions in which bulk InP does not dissolve due to its covalent bond nature, was observed at two steps: phase transfer and shell formation. This dissolution competed with the formation of the ZnS layer at the start of UV irradiation. Since the UV irradiation enables creation of a thick shell by optimizing the synthesizing conditions, high PL efficiency (30-68%) was obtained in water with sufficient stability. This was quantitatively explained by quantum mechanical calculations. The PL decay behavior of these water-soluble InP/ZnS NCs did not show obvious size-dependence, unlike HF-treated ones. This is attributed to the well-passivated surface states of the NCs due to their thick ZnS shell. The NCs showed a significantly higher In/P ratio than those previously reported. This indicates that In ions were preferentially located on the surface of the InP core in the NCs.

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We have controlled in-plane stress in the CoO2 sublattice of (Ca2CoO3)xCoO2 by cationic substitution of alkaline earth metals at the Ca site. The effects of the magnitude of the stress on thermoelectric properties have been investigated by using highly grain-oriented polycrystalline thin films. The in-plane compressive stress is released by the partial substitution of Sr for Ca. With increasing in-plane stress, the magnitude of thermoelectric power and resistivity increased.