Lower bounds on the cardinality of the maximum matchings of planar graphs

Lower bounds on the cardinality of the maximum matchings of planar graphs, with a constraint on the minimum degree, are established in terms of a linear polynomial of the number of vertices. The bounds depend upon the minimum degree and the connectivity of graphs. Some examples are given which show that all the lower bounds are best possible in the sense that neither the coefficients nor the constant terms can be improved.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field.     

Synthesis and properties of polydiene and polypentenamer having formyl or hydroxymethyl groups and their photosensitive derivatives

cis-1,4-Polybutadiene and polypentenamer having pendant functional groups such as formyl, aldoxime, hydroxymethyl, or cinnamoyloxymethyl groups have been prepared, and some of their properties were investigated in terms of structural effect on physical properties of these polymers. cis-1, 4-Polybutadiene and polypentenamer having a different content of formyl group were prepared by the hydroformylation reaction with rhodium catalyst under mild conditions. The pendant formyl group was reduced to a hydroxymethyl group by using various reducing agents such as sodium borohydride or sodium trimethoxyborohydride which were effective to avoid a crosslinking reaction among the formyl groups. Glass transition temperature of polypentenamer having hydroxymethyl groups increased with increasing the content of the hydroxymethyl groups in the polymer. Cinnamoyl group was introduced into the polypentenamer having hydroxymethyl groups by reacting with cinnamoyl chloride so as to prepare a photosensitive rubber. The relationships between the photosensitivity of the cinnamoylated polypentenamer and the mobility of polymer main chains have been elucidated. A linear relationship between the photodimerization rate constant and the difference between ultraviolet (UV) irradiation temperature and the glass transition temperature of the polymer was found. It has become apparent that the photosensitivity of cinnamoylated polypentenamer can be estimated by the glass transition temperature of the original polymer, UV irradiation temperature, activation energy of the dimerization, and γ, which is a coefficient of the relationship between the photosensitive group concentration and the glass transition temperature of the polymer.     

The ring-opening reactions of propylene oxide with chloroacetic acids

The association constants, k_d, of dichloroacetic acid (DCA) with ethers (tetrahydrofuran, tetrahydropyran, dibutyl ether, 1,4-dioxane) were measured in CCl₄ from 20°–40° using near IR spectra. A linear relationship between log₁₀K_D and log₁₀K_p was found, where K_p was the association constant of phenol with ethers. On the basis of their relationship the association constants of DCA with propylene oxide (PO) were estimated from those of phenol with PO. Furthermore, the ring-openings of PO with chloroacetic acids (trichloroacetic acid (TCA) or DCA) were studied kinetically in toluene. It was found that the PO-TCA reactions obeyed the second-order kinetics dependent on the first order both in the PO concn and in the TCA concn, and that the PO-DCA reactions obeyed the third-order kinetics on the first order in the PO concn and on the second order in the DCA concn. The elementary reaction rate constants were estimated on the basis of measured rate constants and estimated equilibrium constants and activation enthalpies were obtained from their temp dependence to be 13.5 $\text{kcal}\text{mol}$ in the PO-TCA reaction and 19.7 $\text{kcal}\text{mol}$ in the PO-DCA reaction. The mechanism was based on the activation enthalpy of elementary reaction and it was concluded that the main factor governing the reaction was the extent of activation of the protonated oxirane (the extent of polarization of the C-O bond in the oxirane ring).     

A novel cyclodimerization of 3-methyl-3-butenenitrile. An efficient synthesis of 4-cyanoisophorone

Dimerization of 3-methyl-3-butenenitrile (1) by lithium dialkylamide gave selectively 3,5,5-trimethyl-1-amino-4-cyanocyclohexa-1,3-diene (2), which was easily hydrolyzed by aqueous acid to afford 4-cyanoisophorone (3,5,5-trimethyl-4-cyano-cyclohex-2-enone) (3) quantitatively.     

Class field theory for a product of curves over a local field

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K ₁ of a product of curves over a number field.      

Volume collapsed three-manifolds with a lower curvature bound

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.     

An NC Parallel Algorithm for Edge-Coloring Series–Parallel Multigraphs

Many combinatorial problems can be efficiently solved in parallel for series–parallel multigraphs. The edge-coloring problem is one of a few combinatorial problems for which no NC parallel algorithm has been obtained for series–parallel multigraphs. This paper gives an NC parallel algorithm for the problem on series–parallel multigraphsG. It takesO(log n) time withO(Δn/log n) processors, wherenis the number of vertices and Δ is the maximum degree ofG.     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.