Some colouring problems for Paley graphs

The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved

Node-weighted Steiner tree and group Steiner tree in planar graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group

Modelling of an imaging beamline at the ISIS pulsed neutron source

A combined neutron imaging and neutron diffraction facility, IMAT, is currently being built at the pulsed neutron spallation source ISIS in the U.K. A supermirror neutron guide is required to combine imaging and diffraction modes at the sample position in order to obtain suitable time of flight resolutions for energy selective imaging and diffraction experiments. IMAT will make use of a straight neutron guide and we consider here the optimization of the supermirror guide dimensions and characterisation of the resulting beam characteristics, including the homogeneity of the flux distribution in space and energy and the average and peak neutron fluxes. These investigations take into account some main design criteria: to maximise the neutron flux, to minimise geometrical artefacts in the open beam image at the sample position and to obtain a good energy resolution whilst retaining a large neutron bandwidth. All of these are desirable beam characteristics for the proposed imaging and diffraction analysis modes of IMAT