A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

Two-wavelength and power-balanced oscillation of a CO2 laser for application to differential absorption measurements

A new technique of alternate two-wavelength oscillation of a CO₂ laser is discussed for application to various differential absorption spectroscopic measurements. Power-balanced, two-adjacent branch oscillation using a single CO₂ laser was achieved by modulating the angle of a mirror inside the laser cavity and adjusting automatically the cavity length. The two-wavelength modulation frequency was extended up to about 1.2 kHz. Line tuning and power modulation characteristics were studied. The laser was used in long-path differential absorption measurements of ethylene air pollution molecules to demonstrate the capability of this power-balanced, two-wavelength oscillation method. The minimum detectable absorption was nearly 1×10^–3 in a short-path cell experiment and 3×10^–3 in a long-path experiment.     

Irreversible circulation and the undamped spiking in lasers

Our new theory of fluctuation was applied to the undamped spiking in lasers with single mode saturable absorption. At the marginal points, which lead to limit cycles, it was found that the irreversible circulation α and the residual circulation α′ diverge together with the variance σ. The model system of this article has effectively three degrees of freedom; however, it was demonstrated that the dominant motion and the fluctuation have two dimensional character in the neighbourhood of a hard mode instability.     

On the maximum matchings of regular multigraphs

The lower bounds on the cardinality of the maximum matchings of regular multigraphs are established in terms of the number of vertices, the degree of vertices and the edge-connectivity of a multigraph. The bounds are attained by infinitely many multigraphs, so are best possible.     

WCl6/LiAlH4 Promoted transformation of imines into secondary and tertiary amines

The reaction of WCl₆/LiAlH₄ with imines, R′NCHR, gave tertiary amines, R′N(CH₂R)₂, and secondary amines, R′NHCHRCH₂R. Isotope labeling experiments revealed that the reaction involved two types of azatungstenacyclobutanes, $\text{WNR′CHRC}$HR and $\text{WCHRNR′C}$HR, produced from the reaction of an alkylidene tungsten intermediate with the imine CN double bond. Formation of these metallacyclobutanes is highly dependent on the solvent used.     

Lower bounds for combinatorial problems on graphs

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is established in a unified way for a rather large class of problems.     

A linear 5-coloring algorithm of planar graphs

A simple linear algorithm is presented for coloring planar graphs with at most five colors. The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.