Parameterized Complexity of Equitable Coloring

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph

Plastic Deformation of 2D Crumpled Wires

When a single long piece of elastic wire is injected trough channels into a confining two-dimensional cavity, a complex structure of hierarchical loops is formed. In the limit of maximum packing density, these structures are described by several scaling laws. In this paper it is investigated this packing process but using plastic wires which give origin to completely irreversible structures of different morphology. In particular, it is studied experimentally the plastic deformation from circular to oblate configurations of crumpled wires, obtained by the application of an axial strain. Among other things, it is shown that in spite of plasticity, irreversibility, and very large deformations, scaling is still observed.Comment: 5 pages, 6 figure

Polynomial Realization of sℓq(2)s\ell_q(2) and Fusion Rules at Exceptional Values of qq

Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated

Produtividade do araçá-vermelho (Psidium cattleyanum L.) em pomar de seis anos sob manejo ecológico.

Editado por Ivan Rodrigues de Almeida, Leonardo Ferreira Dutra e Jamir Luis Silva da Silva. 1 CD-ROM

Dynamics and control of measles in Portugal: Accessing the impact of anticipating the age for the first dose of MMR from 15 to 12 months of age

The all-time low incidence of measles in Portugal in the recent years, raises questions regarding whether the disease has been eliminated, the role of recent control measures, and the epidemiological consequences of the rise in the proportion of newborns to vaccinated mothers, as opposed to those born to mothers who acquired immunity by natural infection. We estimate the vaccination coverage against measles in Portugal. on a cohort-by-cohort basis, and incorporate this information into an age-structured seasonally-driven mathematical model aimed at reproducing measles dynamics in the past decades. The model reproduces documented trends in disease notifications and the serological profile of the Portuguese population, as estimated by a recent National Serological, Survey. We provide evidence that the effective reproduction number (R-e) of measles has been driven below 1 in Portugal, and that sustained measles elimination is crucially dependent upon the maintenance of a high (>95%) coverage with the MMR I vaccine in the future. If the vaccination coverage decreases to levels around 90% the anticipation of the first dose of the MMR I from 15 to 12 months of age, will. ensure that R-e remains below 1. (C) 2008 Elsevier Ltd. All. rights reserve