Three-coloring triangle-free graphs on surfaces III. Graphs of girth five

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark

Twin-width of graphs on surfaces

Twin-width is a width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$

Strong modeling limits of graphs with bounded tree-width

The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Ne\v{s}et\v{r}il and Ossona de Mendez [J. Symbolic Logic 84 (2019), 452--472] proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Ne\v{s}et\v{r}il and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52] and for first order sequences of graphs with bounded path-width by Gajarsk\'y et al. [Random Structures Algorithms 50 (2017), 612--635]. We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.Comment: arXiv admin note: text overlap with arXiv:1504.0812

Fractional colorings of cubic graphs with large girth

International audienceWe show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978, which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid for random cubic graphs as well, as it improves existing lower bounds on the maximum cut in cubic graphs with large girth

Nutrigenomics analyze of expression of extracellular leptin receptor by the following essential oil monitoring at the avian models

Leptin gene was identified in 1994 by positional cloning. His mutation is considered extreme obesity surface phenotype and infertility in ob/ob mice. Most of the research, which followed the discovery of this hormone, focused on the role of leptin in regulating body weight,  in order to clarify the pathophysiology of obesity. Many research results show that leptin is not only important in regulating food intake and energy balance, but also performs functions such as metabolic and neuroendocrine hormone. Using herbs and essential oils depends on their antimicrobial activity. Most plants have favorable multifunctional properties, which are the specific content of bioactive components. Some authors characterize fytogénne substance such as natural substancese plant origin, which leave no residues in animal products and is not necessary to keep the trade period before slaughter animals. Analyses suggest that the structural function of the receptor exists as a dimer constructively in the plasma membrane. Each receptor dimer pair is reversibly bound to one molecule of leptin. When bound, signaling pathways are responsible for beginning the activation receptor associated Janus kinase 2 (JAK2) and tyrosine phosphorylation of two key residues in the intracellular part of receptor

Density maximizers of layered permutations

A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if σ is a layered permutation, then the density of σ in a permutation of order n is maximized by a layered permutation. Albert, Atkinson, Handley, Holtonand Stromquist [Electron. J. Combin. 9 (2002), #R5] claimed that the density of a layered permutation with layers of sizes (a,1,b) where a,b > 2 is asymptotically maximized by layered permutations with a bounded number of layers, and conjectured that the same holds if a layered permutation has no consecutive layers of size one and its first and last layers are of size at least two. We show that, if σ is a layered permutation whose first layer is sufficiently large and second layer is of size one, then the number of layers tends to infinity in every sequence of layered permutations asymptotically maximizing the density of σ. This disproves the conjecture and the claim of Albert et al. We complement this result by giving sufficient conditions on a layered permutation to have asymptotic or exact maximizers with a bounded number of layers

Eo-Alpine metamorphism and the ‘mid-Miocene thermal event’ in the Western Carpathians (Slovakia): New evidence from multiple thermochronology

A combination of zircon (U–Th)/He (ZHe), apatite fission track (AFT) and apatite (U–Th)/He (AHe) dating methods is applied to constrain the metamorphic and exhumation history of the Tatric part of the Branisko Mountains in the Western Carpathians. ZHe ages from the basement samples prove the basement experienced a very low-grade to low-grade Eo-Alpine metamorphic overprint in mid-Cretaceous times. Miocene AFT and AHe ages found in the basement and in the Palaeogene sediments conclusively demonstrate that the Branisko Mts experienced a ‘mid-Miocene thermal event’. This thermal event had a regional character and was related to magmatic and/or burial heating that exposed the sediment and basement samples to ~ 120–130°C and ~ 100–190°C, respectively

Multiple low-temperature thermochronology constraints on exhumation of the Tatra Mountains: New implication for the complex evolution of the Western Carpathians in the Cenozoic

The tectonothermal evolution of the highest mountain range in the Carpathian arc—the Tatra Mountains— is investigated by zircon and apatite fission track and zircon (U-Th)/He (ZHe) dating methods in order to unravel the disputed exhumation and geodynamic processes in the Western Carpathians. Our data in combination with geological evidences reveal a complex Cenozoic history, with four major tectonothermal events: (i) a very low grade metamorphism of the crystalline basement at temperatures >240°C due to tectonic burial during the Eo-Alpine collision in the Late Cretaceous (~80 Ma); (ii) exhumation and cooling of the basement to temperatures 150°C after burial to 5–9 km depths by the Paleogene fore-arc basin; (iv) final exhumation of the segmented basement blocks during Oligocene-Miocene (32–11 Ma) owing to lateral extrusion of the North Pannonian plate and its collision with the European foreland. The spatial pattern of thermochronological data suggests asymmetric exhumation of the Tatra Mountains, beginning in the northwest at ~30–20 Ma with low cooling rates (~1–5°C/Ma) and propagating toward the major fault bounding the range in the south, where the youngest cooling ages (16–9 Ma) and fastest cooling rates (~10–20°C/Ma) are found. Our data prove that the Tatra Mountains shared Cenozoic evolution of other crystalline core mountains in the Western Carpathians. However, the Miocene ZHe ages suggest that the Tatra Mountains were buried to the greatest depths in the Paleogene-Early Miocene and experienced the greatest amount of Miocene exhumation