1,040 research outputs found

    Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

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    A kk-bisection of a bridgeless cubic graph GG is a 22-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most kk. Ban and Linial conjectured that every bridgeless cubic graph admits a 22-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph GG with E(G)0(mod2)|E(G)| \equiv 0 \pmod 2 has a 22-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying 22-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

    Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries

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    Using exact computations we study the classical hard-core monomer-dimer models on m x n plane lattice strips with free boundaries. For an arbitrary number v of monomers (or vacancies), we found a logarithmic correction term in the finite-size correction of the free energy. The coefficient of the logarithmic correction term depends on the number of monomers present (v) and the parity of the width n of the lattice strip: the coefficient equals to v when n is odd, and v/2 when n is even. The results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers.Comment: 4 pages, 2 figure

    Parameterized Study of the Test Cover Problem

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    We carry out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity. In the {\sc Test Cover} problem we are given a set [n]={1,...,n}[n]=\{1,...,n\} of items together with a collection, T\cal T, of distinct subsets of these items called tests. We assume that T\cal T is a test cover, i.e., for each pair of items there is a test in T\cal T containing exactly one of these items. The objective is to find a minimum size subcollection of T\cal T, which is still a test cover. The generic parameterized version of {\sc Test Cover} is denoted by p(k,n,T)p(k,n,|{\cal T}|)-{\sc Test Cover}. Here, we are given ([n],T)([n],\cal{T}) and a positive integer parameter kk as input and the objective is to decide whether there is a test cover of size at most p(k,n,T)p(k,n,|{\cal T}|). We study four parameterizations for {\sc Test Cover} and obtain the following: (a) kk-{\sc Test Cover}, and (nk)(n-k)-{\sc Test Cover} are fixed-parameter tractable (FPT). (b) (Tk)(|{\cal T}|-k)-{\sc Test Cover} and (logn+k)(\log n+k)-{\sc Test Cover} are W[1]-hard. Thus, it is unlikely that these problems are FPT

    Association study of suicidal behavior and affective disorders with a genetic polymorphism in ABCG1, a positional candidate on chromosome 21q22.3

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    The gene that codes for the ABC transporter ABCG1 is located in a chromosomal susceptibility region (21q22.3) for affective disorders. Genetic variations in ABCG1 have been associated with affective disorders in Japanese males. In this study, we investigated the distribution of a G2457A polymorphism in patients with affective disorders, suicide attempters with various psychiatric diagnoses and healthy subjects, We initially found a trend towards a modest association with affective disorders in males (p = 0.046 for allele frequencies and p = 0.046 for AA versus GG). We conducted a replication study with independent patients and controls, There was no association with affective disorders, either in the replication or in the combined group, Furthermore, we found no association with suicidal behavior, These findings do not support the hypothesis that ABCG1 is a susceptibility gene for affective disorders or suicidal behavior. Copyright (C) 2000 S. Karger AG, Basel

    Fundamental Cycles and Graph Embeddings

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    In this paper we present a new Good Characterization of maximum genus of a graph which makes a common generalization of the works of Xuong, Liu, and Fu et al. Based on this, we find a new polynomially bounded algorithm to find the maximum genus of a graph

    The Discovery of Cherenkov Radiation and its use in the detection of extensive air showers

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    Cascades of charged particles are created when high-energy cosmic rays enter the earth's atmosphere: these 'extensive air-showers' are studied to gain information on the energy spectrum, arrival direction distribution and mass composition of the particles above 1014 eV where direct observations using instruments carried by balloons or satellites become impractical. Detection of light in the visible and ultra-violet ranges of the electromagnetic spectrum plays a key role in this work, the two processes involved being the emission of Cherenkov light and the production of fluorescence radiation. In this paper I will outline some of the history of the discovery of the Cherenkov process and describe the use to which it has been put in the study of extensive air-showers at ground level.Comment: To appear in Proceedings of CRIS2010: Cosmic Ray International Seminar on '100 years of Cosmic Rays: from Pioneering Experiments to Physics in Space
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