Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings

Long cycles in graphs containing a 2-factor with many odd components

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian

Pancyclicity of Hamiltonian line graphs

Let f(n) be the smallest integer such that for every graph G of order n with minimum degree 3(G)>f(n), the line graph L(G) of G is pancyclic whenever L(G) is hamiltonian. Results are proved showing that f(n) = ®(n 1/3)

Formation of Double Neutron Stars, Millisecond Pulsars and Double Black Holes

The 1982 model for the formation of the Hulse-Taylor binary radio pulsar PSR B1913+16 is described, which since has become the standard model for the formation of double neutron stars, confirmed by the 2003 discovery of the double pulsar system PSR J0737-3039AB. A brief overview is given of the present status of our knowledge of the double neutron stars, of which 15 systems are presently known. The binary-recycling model for the formation of millisecond pulsars is described, as put forward independently by Alpar et al. (1982), Radhakrishnan and Srinivasan (1982) and Fabian et al. (1983). This now is the standard model for the formation of these objects, confirmed by the discovery in 1998 of the accreting millisecond X-ray pulsars. It is noticed that the formation process of close double black holes has analogies to that of close double neutron stars, extended to binaries of larger iinitial component masses, although there are also considerable differences in the physics of the binary evolution at these larger masses.Comment: Has appeared in Journal of Astrophysics and Astronomy special issue on 'Physics of Neutron Stars and Related Objects', celebrating the 75th birth year of G. Srinivasa

Does bank capital matter for monetary transmission?

Paper for a conference sponsored by the Federal Reserve Bank of New York entitled Financial Innovation and Monetary TransmissionBank capital ; Monetary policy

A generalization of Ore's Theorem involving neighborhood unions

AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2

Long cycles, degree sums and neighborhood unions

AbstractFor a graph G, define the parameters α(G)=max{|S| |S is an independent set of vertices of G}, σk(G)=min{∑ki=1d(vi)|{v1,…,vk} is an independent set} and NCk(G)= min{|∪ki=1 N(vi)∥{v1,…,vk} is an independent set} (k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,n+NCr+5+∈(n+r)(G)-α(G)}, where ε(i)=3(⌈13i⌉−13i). This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,2NC⌊18(n+6r+17)⌋(G)}. Analogous results are established for 2-connected graphs

A catalogue of low-mass X-ray binaries in the Galaxy, LMC, and SMC (Fourth edition)

We present a new edition of the catalogue of the low-mass X-ray binaries in the Galaxy and the Magellanic Clouds. The catalogue contains source name(s), coordinates, finding chart, X-ray luminosity, system parameters, and stellar parameters of the components and other characteristic properties of 187 low-mass X-ray binaries, together with a comprehensive selection of the relevant literature. The aim of this catalogue is to provide the reader with some basic information on the X-ray sources and their counterparts in other wavelength ranges ($\gamma$-rays, UV, optical, IR, and radio). Some sources, however, are only tentatively identified as low-mass X-ray binaries on the basis of their X-ray properties similar to the known low-mass X-ray binaries. Further identification in other wavelength bands is needed to finally determine the nature of these sources. In cases where there is some doubt about the low-mass nature of the X-ray binary this is mentioned. Literature published before 1 October 2006 has, as far as possible, been taken into account.Comment: 45 pages, catalogue include