Coloring a graph optimally with two colors

AbstractLet G be a graph with point set V. A (2-)coloring of G is a map of V to {red, white}. An error occurs whenever the two endpoints of a line have the same color. An optimal coloring of G is a coloring of G for which the number of errors is minimum. The minimum number of errors is denoted by γ(G), we derive upper and lower bounds for γ(G) and prove that if G is a graph with n points and m lines, then max{0, m−⌊14n2⌋}⩽γ(G)⩽⌊12m−14(h(m)−1)⌋, where h(m)=min{n¦m⩽(n2)}. The lower bound is sharp, and for infinitely many values of m the upper bound is attained for all sufficiently large n

Bipartite regular graphs with fixed diameter

For given nonnegative integers k and D, we consider the problem of determining n,(k, D) , the smallest number n for which there exists a k-regular bipartite graph on n vertices with diameter D. We solve the problem for all pairs ( k, D) with D + 2 (mod 4) and D + 3 (mod 4), for all pairs ( k, D) with k even or k prime and D + 3 (mod 4), for all pairs with D I 9 or k I 4, and for a few other pairs. In the remaining cases, we obtain lower and upper bounds for n,,(k, D). 0 7995 John Wdey & Sons, /nc. 1

Amsterdam

Contains fulltext : 127768.pdf (publisher's version ) (Open Access

Improving the performance of periodic real-time processes: a graph theoretical approach

In this paper the performance gain obtained by combining parallel periodic real-time processes is elaborated. In certain single-core mono-processor configurations, for example embedded control systems in robotics comprising many short processes, process context switches may consume a considerable amount of the available processing power. For this reason it can be advantageous to combine processes, to reduce the number of context switches and thereby increase the performance of the application. As we consider robotic applications only, often consisting of processes with identical periods, release times and deadlines, we restrict these configurations to periodic real-time processes executing on a single-core mono-processor. By graph theoretical concepts and means, we provide necessary and sufficient conditions so that the number of context switches can be reduced by combining synchronising processes

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

On a Directed Tree Problem Motivated by a Newly Introduced Graph Product

In this paper we introduce and study a directed tree problem motivated by a new graph product that we have recently introduced and analysed in two conference contributions in the context of periodic real-time processes. While the two conference papers were focussing more on the applications, here we mainly deal with the graph theoretical and computational complexity issues. We show that the directed tree problem is NP-complete and present and compare several heuristics for this problem

Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale

The detection of sound begins when energy derived from acoustic stimuli deflects the hair bundles atop hair cells. As hair bundles move, the viscous friction between stereocilia and the surrounding liquid poses a fundamental challenge to the ear's high sensitivity and sharp frequency selectivity. Part of the solution to this problem lies in the active process that uses energy for frequency-selective sound amplification. Here we demonstrate that a complementary part involves the fluid-structure interaction between the liquid within the hair bundle and the stereocilia. Using force measurement on a dynamically scaled model, finite-element analysis, analytical estimation of hydrodynamic forces, stochastic simulation and high-resolution interferometric measurement of hair bundles, we characterize the origin and magnitude of the forces between individual stereocilia during small hair-bundle deflections. We find that the close apposition of stereocilia effectively immobilizes the liquid between them, which reduces the drag and suppresses the relative squeezing but not the sliding mode of stereociliary motion. The obliquely oriented tip links couple the mechanotransduction channels to this least dissipative coherent mode, whereas the elastic horizontal top connectors stabilize the structure, further reducing the drag. As measured from the distortion products associated with channel gating at physiological stimulation amplitudes of tens of nanometres, the balance of forces in a hair bundle permits a relative mode of motion between adjacent stereocilia that encompasses only a fraction of a nanometre. A combination of high-resolution experiments and detailed numerical modelling of fluid-structure interactions reveals the physical principles behind the basic structural features of hair bundles and shows quantitatively how these organelles are adapted to the needs of sensitive mechanotransduction.Comment: 21 pages, including 3 figures. For supplementary information, please see the online version of the article at http://www.nature.com/natur

Computing with carbon nanotubes: optimization of threshold logic gates using disordered nanotube/polymer composites

This paper explores the use of single-walled carbon nanotube (SWCNT)/poly(butyl methacrylate) composites as a material for use in unconventional computing. The mechanical and electrical properties of the materials are investigated. The resulting data reveal a correlation between the SWCNT concentration/viscosity/conductivity and the computational capability of the composite. The viscosity increases significantly with the addition of SWCNTs to the polymer, mechanically reinforcing the host material and changing the electrical properties of the composite. The electrical conduction is found to depend strongly on the nanotube concentration; Poole-Frenkel conduction appears to dominate the conductivity at very low concentrations (0.11% by weight). The viscosity and conductivity both show a threshold point around 1% SWCNT concentration; this value is shown to be related to the computational performance of the material. A simple optimization of threshold logic gates shows that satisfactory computation is only achieved above a SWCNT concentration of 1%. In addition, there is some evidence that further above this threshold the computational efficiency begins to decrease