Binding Number,Minimum Degree,and Cycle Structure in Graphs

A well‐known theorem of Woodall states that if a graph G has binding number at least 3/2, then G is hamiltonian. We generalize Woodall's theorem as follows.     

Gray codes extending quadratic matchings

Is it true that every matching in the n-dimensional hypercube can be extended to a Gray code? More than two decades have passed since Ruskey and Savage asked this question and the problem still remains open. A solution is known only in some special cases, including perfect matchings or matchings of linear size. This article shows that the answer to the Ruskey–Savage problem is affirmative for every matching of size at most . The proof is based on an inductive construction that extends balanced matchings in the completion of the hypercube by edges of into a Hamilton cycle of . On the other hand, we show that for every there is a balanced matching in of size that cannot be extended in this way.     

Chemists around the World,Take Your Part in the Circular Economy!

Based on recent examples and initiatives reported in the literature, this concept article discusses how chemistry can contribute to the circular economy approach in order to improve our current and future economical, societal, and environmental system. Through five proposed levels of contribution, chemists can take a significant part in this global approach via the consideration of green chemistry principles, the simplification of syntheses, the limitation of complex products preparation, the efficient utilization of resources but also the novel ways of waste valorization. A more systematic and generalized environmental and economic assessment from the lab-scale is also recommended. At last, chemists have to work even more collaboratively and in a multidisciplinary way, within chemistry and beyond.     

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A -bisection of a bridgeless cubic graph is a -colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most . Ban and Linial Conjectured that every bridgeless cubic graph admits a -bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph with has a -edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban-Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above-mentioned conjectures. Moreover, we prove Ban-Linial's Conjecture for cubic-cycle permutation graphs. As a by-product of studying -edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.     

Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube $Q_n$ for $n\ge 2$ can be extended to a hamiltonian cycle of $Q_n$. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of $Q_n$ for $n\ge 2$ can be extended to two or more hamiltonian cycles of $Q_n$. In this paper, we prove that every perfect matching of $Q_n$ for $n\ge 4$ can be extended to at least $2^{2^{n?4}}$ different hamiltonian cycles of $Q_n$.     

Accurate prediction of norbornadiene cycle enthalpies by DLPNO-CCSD(T1)/CBS method

The DLPNO-CCSD(T₁)/CBS method combined with simple reactions containing small reference species leads to an improvement in the accuracy of theoretically evaluated enthalpies of formation of medium-sized polyalicyclic hydrocarbons when compared with the widely used composite approach. The efficiency of the DLPNO-CCSD(T₁)/CBS method is most vividly demonstrated by comparing with the results of G4 calculations for adamantane. The most important factor in choosing appropriate working reaction is the same number of species on both sides of the equation. Among these reactions, the reactions with small enthalpy change usually provide a better cancellation of errors. The DLPNO-CCSD(T₁)/CBS method was used to calculate the enthalpies of formation of compounds belonging to the norbornadiene cycle (norbornadiene, quadricyclane, norbornene, nortricyclane, and norbornane). The most reliable experimental enthalpies of formation are recommended for these compounds by comparing calculated values with conflicting experimental data.     

Characterization of the Insoluble Sludge from the Dissolution of Irradiated Fast Breeder Reactor Fuel

Insoluble sludge is generated in the reprocessing of spent fuel. The sludge obtained from the dissolution of irradiated fuel from the “Joyo” experimental fast reactor was analyzed to evaluate its chemical form. The sludge was collected by the filtration of the dissolved fuel solution, and then washed in nitric acid. The yields of the sludge weight were less than 1% of the total fuel weight. The chemical composition of the sludge was analyzed after decomposition by alkaline fusion. Molybdenum, technetium, ruthenium, rhodium, and palladium were found to be the main constituent elements of the sludge. X-ray diffraction patterns of the sludge were attributable to Mo₄Ru₄RhPd, regardless of the experimental conditions. The concentrations of molybdenum and zirconium in the dissolved fast reactor fuel solutions were low, indicating that zirconium molybdate hydrate is produced in negligible amounts in the process.     

On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set $\{a,b,c,d,e,f\}$ and edge set $\{\{a,b,c\},\{b,c,d\},\{c,d,e\},\{d,e,f\},\{e,f,a\},\{f,a,b\}\}$. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if $v\equiv 1$, 2, 10, 20, 28, or $29(\mathrm{mod}36)$.