Gold and Carbene Relay Catalytic Enantioselective Cycloisomerization/Cyclization Reactions of Ynamides and Enals

The combined use of gold as transition metal catalyst and N‐heterocyclic carbene (NHC) as organic catalyst in the same solution for relay catalytic reactions was disclosed. The ynamide substrate was activated by gold catalyst to form unsaturated ketimine intermediate that subsequently reacted with the enals (via azolium enolate intermediate generated with NHC) effectively to form bicyclic lactam products with excellent diastereo‐ and enantio‐selectivities. The gold and NHC coordination and dissociation can be dynamic and tunable events, and thus allow the co‐existence of both active metal and carbene organic catalysts in appreciable concentrations, for the dual catalytic reaction to proceed.     

Quantum gauge fields and flat connections in 2-dimensional BF theory

The 2-dimensional BF theory is both a gauge theory and a topological Poisson σ-model corresponding to a linear Poisson bracket. In [3], Torossian discovered a connection which governs correlation functions of the BF theory with sources for the B-field. This connection is flat, and it is a close relative of the KZ connection in the WZW model. In this Letter, we show that flatness of the Torossian connection follows from (properly regularized) quantum equations of motion of the BF theory.     

A Solvent Effect in the Reaction of Diazomethane with Norbornane-2,3-dione 3-Hemiketals

A solvent effect in the reaction of diazomethane with norbornane-2,3-dione 3-hemiketal in various aprotic solvents such as ether, dichloromethane, tetrahydrofuran (THF), dioxane, and chloroform is reported. Unlike in methanol, preparatively useful quantities of oxetane derivatives were obtained along with a novel hexahydro-1H-cyclopenta[c]furan-1-one in these aprotic solvents. Dichloromethane and THF gave optimal yields of the two products. In some cases, α-ketoketals were formed as minor products along with the aforementioned products. Interestingly, TMSCHN₂ furnished hexahydro-1H-cyclopenta[c]furan-1-one as the predominant product.     

Infinite Hamilton cycles in squares of locally finite graphs

We prove Diestel's conjecture that the square G² of a 2-connected locally finite graph G has a Hamilton circle, a homeomorphic copy of the complex unit circle S¹ in the Freudenthal compactification of G².     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

The lollipop graph is determined by its Q-spectrum

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q, any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by H_n,p, is obtained by appending a cycle C_p to a pendant vertex of a path P_n−p. In this paper, it is proved that all lollipop graphs are determined by their Q-spectra.     

On-line arbitrarily vertex decomposable suns

We give a complete characterization of on-line arbitrarily vertex decomposable graphs in the family of unicycle graphs called suns. A sun is a graph with maximum degree three, such that deleting vertices of degree one results in a cycle. This result has already been used in another paper to prove some Ore-type conditions for on-line arbitrarily decomposable graphs.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.     

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem [4] states that every planar triangle-free graph is 3-colorable. However, Voigt [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.     

Super edge-connectivity of mixed Cayley graph

A graph X is max-λ if λ(X)=δ(X). A graph X is super-λ if X is max-λ and every minimum edge-cut set of X isolates one vertex. In this paper, we proved that for all but a few exceptions, the mixed Cayley graph which is defined as a new kind of semi-regular graph is max-λ and super-λ.