Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.     

Time-Ordering and a Generalized Magnus Expansion

Both the classical time-ordering and the Magnus expansion are well known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including, however, directly a non-trivial initial condition. As a by-product we recover a variant of the time-ordering operation, known as ${\mathsf{T}^\ast}$ -ordering. Eventually, placing our results in the general context of Rota–Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations.     

Possible inertias for the hermitian part of A + BX

Given AεM_n (C) and BεM _n,k (C) all possible inertias occurring in the Hermitian part of A+BX are determined as X runs over M_k,n(C).     

Packing Triangles in Regular Tournaments

We prove that a regular tournament with n vertices has more than pairwise arc‐disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than , so these tournaments do not have pairwise arc‐disjoint triangles.     

Spontaneous Resolution by Crystallization of an Octanuclear Iron(III) Complex Using Only Racemic Reagents

The P and M enantiomers of the octanuclear [Fe₈(μ₄‐O)₄(μ‐4‐Cl‐pz)₁₂Cl₄] complex, having T symmetry, were resolved by temporary substitution of chloride ligands by racemic 4‐^sBu‐phenolates and subsequent crystallization, where the (S)‐ and (R)‐phenolates coordinate selectively to the M and P complexes, respectively. The complexes were characterized by circular dichroism analysis and X‐ray structure determination. This work constitutes a rare example of enantiomeric recognition resulting in spontaneous resolution upon crystallization.     

Stable,Reactive, and Orthogonal Tetrazines: Dispersion Forces Promote the Cycloaddition with Isonitriles

The isocyano group is a structurally compact bioorthogonal functional group that reacts with tetrazines under physiological conditions. Now it is shown that bulky tetrazine substituents accelerate this cycloaddition. Computational studies suggest that dispersion forces between the isocyano group and the tetrazine substituents in the transition state contribute to the atypical structure–activity relationship. Stable asymmetric tetrazines that react with isonitriles at rate constants as high as 57 L mol^?1 s^?1 were accessible by combining bulky and electron‐withdrawing substituents. Sterically encumbered tetrazines react selectively with isonitriles in the presence of strained alkenes/alkynes, which allows for the orthogonal labeling of three proteins. The established principles will open new opportunities for developing tetrazine reactants with improved characteristics for diverse labeling and release applications with isonitriles.     

Direct Formic Acid Mediated Z‐Selective Reductive Coupling of Dienes and Aldehydes

Methods for the addition of unsaturated nucleophiles to carbonyls to generate Z‐olefin products remain rare and often require either alkyl borane or zinc reductants, limiting their utility. Demonstrated here is that formic acid mediates the Rh‐catalyzed, Z‐selective coupling of dienes and aldehydes. The process is distinguished by broad tolerance towards reducible or electrophilic groups. Kinetic analysis suggests that generation of the catalytically active Rh intermediate by ligand dissociation is the rate‐determining step. The rapid generation and trapping of Rh‐allyl intermediates is key to preventing chain‐walking isomerization events that plague related protocols. Insights gained through this study may have wider implications in selective metal‐catalyzed hydrofunctionalization reactions.