Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

Carboniferous and Permian Rugosochonetidae (Brachiopoda) from West Spitsbergen

The rugosochonetid brachiopod species Lissochonetes geinitzianus from the Kazimovian of the Nordenskioldbreen Formation, and Dyoros (Dyoros) mucronata sp. nov., Dyoros (Dyoros) spitzbergianus and Lissochonetes superba from the Artinskian to latest Permian Kapp Starostin Formation in West Spitsbergen are described and figured. Dyoros is generally restricted to the Boreal Realm, whereas Lissochonetes is mostly distributed in the Boreal Realm, but occasionally present in the Palaeoequatorial and Gondwanan Realms<br /

The Holy Cross Mountains (Poland) terranes palaeoposition and depositional environment in Silurian : New insights from rock magnetic studies

The Holy Cross Mountains (HCM) in Poland, is an isolated natural outcrop of Palaeozoic rocks located within the Trans-European Suture Zone, a tectonic collage of continental terranes adjacent to the Tornquist margin of the Baltica. This uniqueness made the HCM a target for palaeogeographic research. Based on the facies differences, the HCM had been divided into two major units, the southern (the Kielce Unit) and northern (the Lysogory Unit) part (SHCM and NHCM, respectively). Their position in relation to each other and the Baltica continent during Silurian times is still a matter of discussion, whether both parts of the HCM were separated terranes located along the Baltica margin or they shared in common palaeogeographic history. Here, we present the results of comprehensive rock magnetic measurements applied as a tool to interpret palaeoenvironmental conditions during deposition and burial and therefore allow discussion about the terranes' relative position. To recognize the magnetic mineral composition and texture of studied Silurian graptolitic shales several rock magnetic measurements were conducted including low-temperature Saturated Isothermal Remanent Magnetization, thermal demagnetization of three-component IRM and hysteresis measurements, as well as anisotropy of magnetic susceptibility (AMS). The sampled rocks come from both units of the HCM. In all analysed samples we found single domain (SD) stoichiometric magnetite of mostly diagenetic (i.e. post-depositional) origin and goethite resulting likely from weathering. In turn, detrital magnetite, even if observed in previously investigated Silurian rocks from the Baltica margin, was not identified in this study, what we attribute to dissolution during diagenesis in the deep-water environment. Solely in the NHCM, SD hematite and maghemite grains were observed, which we interpret as detrital in origin. These grains have been preserved in the suboxic environment of the NHCM sub-basin bottom waters due to their resistance to dissolution in marine waters. Considering the deposition conditions (oxygenation of the near-bottom zone) rather similar for both HCM parts, we associate the presence of aeolian hematite grains solely in the NHCM rocks with a more proximal position of the NHCM than the SHCM in relation to the Baltica continent during late Llandovery (Silurian). This conclusion agrees with some existing palaeogeographic models. In addition to petromagnetic studies focused on the analysis of ferromagnets, AMS measurements were also carried out. The results indicate that the magnetic susceptibility is mainly governed by paramagnetic minerals, mostly phyllosilicates with small ferromagnetic contributions. Oblate AMS ellipsoid and distinct bedding parallel foliation indicate prevailing sedimentary-compactional alignment. Observed magnetic lineation of tectonic origin resulting from weak strain is related presumably to Variscian deformations

Faunal dynamics across the Silurian-Devonian positive isotope excursions (delta13C, delta18O) in Podolia, Ukraine: Comparative analysis of the Ireviken and Klonk events

Two global isotopic events, the early Sheinwoodian (early Wenlock) and that at the Silurian–Devonian transition, have been comprehensively studied in representative carbonate successions at Kytayhorod and Dnistrove, respectively, in Podolia, Ukraine, to compare geochemistry and biotic changes related correspondingly to the Ireviken and Klonk events. These two large−scale isotope excursions reveal different regional ecosystem tendencies. The well−defined increasing trend across the Llandovery–Wenlock boundary in siliciclastic input, redox states and, supposedly, bioproductivity, was without strict correlative relations to the major ¹³C enrichment event. The environmental and biotic evolution was forced by eustatic sea−level fluctuations and two−step climate change toward a glaciation episode, but strongly modified by regional epeirogeny movements due to location near the mobile Teisseyre−Törnquist Fault Zone. Thus, the global early Sheinwoodian biogeochemical perturbation was of minor depositional significance in this epeiric sea, as in many other Laurussian domains. Conversely, the Podolian sedimentary record of the Klonk Event exhibits temporal links to the abrupt δ¹³C anomaly, overprinted by a tectonically driven deepening pulse in the crucial S–D boundary interval. This carbon cycling turnover was reflected in the regional carbonate crisis and cooling episodes, paired with a tendency towards eutrophication and recurrent oxygen deficiency, but also with major storms and possible upwelling. Faunal responses in both Podolian sections follow some characters of the Silurian pattern worldwide, as manifested by conodont changeover prior to the major early Sheinwoodian isotopic/climatic anomaly. This contrasts with the relative brachiopod and chitinozoan resistances in the course of the Ireviken Event. Also, during the Klonk Event, a moderate faunal turnover, both in benthic and pelagic groups, occurred only near the very beginning of the prolonged ¹³C−enriched timespan across the system boundary, possibly due to progressive dysoxia and temperature drop. The characters point to a peculiarity of the Klonk Event by comparison with the Silurian global events, and some similarity already to the succeeding Devonian transgressive/anoxic episodes

Codon 129 polymorphism of the human prion protein influences the kinetics of amyloid formation

The human prion protein (PrP) has a common polymorphism at residue 129, which can be valine or methionine. This polymorphism has a strong influence on susceptibility to prion diseases and on prion-strain properties. Previous work has shown that this amino acid variation has no measurable effect on the native structure of cellular PrP (PrPC). Here, it is shown that the polymorphism does not change the efficiency of conversion to the beta-PrP conformation or affect the binding of copper(II) ions. However, in a partially denatured conformation, the polymorphic variation has a profound influence on the ability of the protein to form amyloid fibrils spontaneously