Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively k-dimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.

Effect of dark matter halo on global spiral modes in galaxies

Low surface brightness (LSB) galaxies form a major class of galaxies, and are characterized by low disc surface density and low star formation rate. These are known to be dominated by dark matter halo from the innermost regions. Here, we study the role of the dark matter halo on the grand-design, m = 2, spiral modes in a galactic disc by carrying out a global mode analysis in the WKB approximation. The Bohr-Sommerfeld quantization rule is used to determine how many discrete global spiral modes are permitted. First, a typical superthin, LSB galaxy UGC 7321 is studied by taking only the galactic disc, modelled as a fluid; and then the disc embedded in a dark matter halo. We find that both cases permit the existence of global spiral modes. This is in contrast to earlier results where the inclusion of dark matter halo was shown to nearly fully suppress local, swing-amplified spiral features. Although technically global modes are permitted in the fluid model as shown here, we argue that due to lack of tidal interactions, these are not triggered in LSB galaxies. For comparison, we carried out a similar analysis for the Galaxy, for which the dark matter halo does not dominate in the inner regions. We show that here too the dark matter halo has little effect, hence the disc embedded in a halo is also able to support global modes. The derived pattern speed of the global mode agrees fairly well with the observed value for the Galaxy.

Effect of strain rate response and pin diameter on mechanically mixed layer formation and wear mechanisms in a Ti6Al4V-SS316L pair

Three mechanisms operate during wear of materials. These mechanisms include the Strain Rate Response (SRR - effect of strain rate on plastic deformation), Tribo-Chemical Reactions (TCR) and formation of Mechanically Mixed Layers (MML). The present work investigates the effect of these three in context of the formation of MML. For this wear experiments are done on a pin-on-disc machine using Ti64 as the pin and SS316L as the disc. It is seen that apart from the speed and load, which control the SRR and TCR, the diameter of the pin controls the formation of MML, especially at higher speeds.