On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of $k$ vertex-disjoint cycles in a graph $G$. For an integer $t\ge 1$, let ${\sigma }_t\left(G\right)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We prove that if $G$ has order at least $7k+1$ and ${\sigma }_4\left(G\right)\ge 8k?3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. We also show that the degree sum condition on ${\sigma }_4\left(G\right)$ is sharp and conjecture a degree sum condition on ${\sigma }_t\left(G\right)$ sufficient to imply $G$ contains $k$ vertex-disjoint cycles for $k\ge 2$.     

Cycles with a chord in dense graphs

A cycle of order $k$ is called a $k$-cycle. A non-induced cycle is called a chorded cycle. Let $n$ be an integer with $n\ge 4$. Then a graph $G$ of order $n$ is chorded pancyclic if $G$ contains a chorded $k$-cycle for every integer $k$ with $4\le k\le n$. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order $n$ satisfying ${deg}_{G}u+{deg}_{G}v\ge n$ for every pair of nonadjacent vertices $u$,  $v$ in $G$ is chorded pancyclic unless $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, the Cartesian product of $K_3$ and $K_2$. They also conjectured that if $G$ is Hamiltonian, we can replace the degree sum condition with the weaker density condition $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph $G$ of order $n$ with $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ contains a $k$-cycle, then $G$ contains a chorded $k$-cycle, unless $k=4$ and $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, Then observing that $K_{\frac{n}{2},\frac{n}{2}}$ and $K_3\square K_2$ are exceptions only for $k=4$, we further relax the density condition for sufficiently large $k$.     

Reductions by titanium(II) as catalyzed by titanium(IV)

The cobalt(III) complexes, [(NH3)5CoBr]2+ and [(NH3)5CoI]2+ are reduced by Ti(II) solutions containing Ti(IV), generating nearly linear (zero-order) profiles that become curved only during the last few percent of reaction. Other Co(III)-Ti(II) systems exhibit the usual exponential traces with rates proportional to [Co(III)]. Observed kinetics of the biphasic catalyzed Ti(II)-Co(III)Br and Ti(II)-Co(III)I reactions support the reaction sequence: [Ti(II)(H20)n]2+ + [Ti(IV)F5]- (k1)<==>(k -1) [Ti(II)(H2O)(n-1)]2+ + [(H2O)Ti(IV)F5]-, [Ti(II)(H2O)(n-1)]2+ + Co(III) (k2)--> Ti(III) + Co(II) with rates determined mainly by the slow Ti(IV)-Ti(II) ligand exchange (k1 = 9 x 10(-3) M(-1) s(-1) at 22 degrees C). Computer simulations of the catalyzed Ti(II)-Co(III) reaction in perchlorate-triflate media yield relative rates for reduction by the proposed active [Ti(II)(H2O)(n-1)]2+ intermediate; k(Br)/k(I) = 8.     

On H‐immersions

For a fixed multigraph H, possibly containing loops, with V(H) = {h₁,…, h_k}, we say a graph G is H‐linked if for every choice of k vertices v₁,…,v_k in G, there exists a subdivision of H in G such that v_i represents h_i (for all i). An H‐immersion in G is similar except that the paths in G, playing the role of the edges of H, are only required to be edge disjoint. In this article, we extend the notion of an H‐linked graph by determining minimum degree conditions for a graph G to contain an H‐immersion with a bounded number of vertex repetitions on any choice of k vertices. In particular, we extend results found in [2,3,5]. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 245–254, 2008     

On Super-RS Algebra and Drinfeld Realization of Quantum Affine Superalgebras

We describe the realization of the super-Reshetikhin–Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel and Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super-RS algebra and the Drinfeld current realization of quantum affine superalgebras is established by using the Gauss decomposition technique of Ding and Frenkel. As an application, we obtain Drinfeld realization of quantum affine superalgebra U_q [osp(1|2)⁽¹⁾] and its degeneration – central extended super-Yangian double DY [osp(12)⁽¹⁾].     

Electron transfer. 141. Reactions of indium(I) with transition metal center oxidants

Aqueous solutions of the hypovalent state In(I) reduce a series of complexes of Fe(III) and selected derivatives of Ir(IV), Cr(V), Pt(IV), Ag(III), and Ni(IV). All reactions yield In(III). Ions derived from Fe(III) and Ir(IV) undergo net le- conversions, whereas the other oxidants change by two units. Reductions of Fe(III) are strongly promoted by increasing pH or by adding Cl-, Br-, NCS-, or N3-. Reaction appears to proceed through monoligated complexes, FeIII(Lig-)2+, and the kinetic response to alteration of added ligand indicates initiation mainly via a bridged transition state FeIII-Lig-InI. Oxidations by Fe(CN)6(3-) are exceptionally rapid, those by FeIII(EDTA) are unusually slow, and redox is blocked by addition of excess F-. Reduction of IrCl6(2-) proceeds somewhat more slowly than predicted by the marcus model for outer-sphere reactions. Conversions of the 2e- oxidants are rapid. For these, multistep routes initiated by le- changes are reasonable, but direct 2e-paths involving oxygen transfer (from CrVO) or Cl+ transfer (from PtCl6(2-)) cannot be ruled out. Whether inner-sphere 2e- transactions without transfer of atomic species can be competitive remains an open question.     

Reaction of energetic hydrogen atoms with 1-chloropropane

The reaction of photochemically generated energetic hydrogen atoms with 1-chloropropane, reaction (1), has been examined for translational energies of H* in the range 40 to 110 kJ mol^?1. Integral probabilities for reaction (1) have been determined, and the phenomenological threshold energy is 47 ± 10 kJ mol^?1. The moderating effect of CO₂ on reaction (1) for hydrogen atoms of initial energy 108 kJ mol^?1 has also been studied. © 1993 John Wiley & Sons, Inc.     

Reactions of tris(oxalato)cobaltate(III) with two-electron reductants

The tris(oxalato)cobaltate(III) complex [Co(C(2)O(4))(3)](3-), E(o)(Co)(III/II)=+0.57 V) is readily reduced by the 2e(-) reagents, Sn(II) and Ge(II), in contrast to (NH(3))(5)CoCl(2+) and (NH(3))(5)CoBr(2+), which are unreactive toward these donors. Rates for the oxalato oxidant are only 10(-3)-10(-2) as great as those for vitamin B(12a)(aquacob(III)alamin, E(o)+0.35 V at pH 1), in accord with the suggestion that reductions of corrin-bound cobalt(III) by Sn(II) and Ge(II) occur predominantly through an additional path involving Co(i). Reductions of the oxalato complex by 2e(-) donors are taken to proceed by initial formation of odd-electron intermediates (e.g., Sn(III) and Ge(III)) which react rapidly with Co(III). Such a two-step sequence is in keeping with the observed behavior of the rare reductant, Ti(II), which is found to be oxidized by [Co(C(2)O(4))(3)](3-) more slowly than (independently prepared) Ti(III) under comparable conditions.     

Electron transfer: 150. The reaction of corrin-bound cobalt(III) with indium(I) — an additional mechanistic variation

(Corrin)cobalt(III) is reduced by indium(I) to its Co(II) counterpart (cob(II)alamin, B_12r) at pH 1-2 in aqueous chloride. In these solutions the aqua oxidant (B_12a) is in mobile anation equilibrium with its chloro derivative. At [Cl^–] 0.03 M, consumption of Co(III) is exponential and proceeds by parallel paths involving the aqua- and chloro-substituted oxidants. At [Cl^–] 0.10 M, rates are governed mainly by an preliminary act requiring In(I) and 2 Cl^–, but no Co(III). This initiation step generates a more reactive In(I) species which is taken to result from a slow heterolysis and loss of ligating water from InCl₂ ^–(aq).