Walker-Breaker Games on Gn,pG_{n,p}

The Maker-Breaker connectivity game and Hamilton cycle game belong to the best studied games in positional games theory, including results on biased games, games on random graphs and fast winning strategies. Recently, the Connector-Breaker game variant, in which Connector has to claim edges such that her graph stays connected throughout the game, as well as the Walker-Breaker game variant, in which Walker has to claim her edges according to a walk, have received growing attention. For instance, London and Pluh\'ar studied the threshold bias for the Connector-Breaker connectivity game on a complete graph $K_n$, and showed that there is a big difference between the cases when Maker's bias equals $1$ or $2$. Moreover, a recent result by the first and third author as well as Kirsch shows that the threshold probability $p$ for the $(2:2)$ Connector-Breaker connectivity game on a random graph $G\sim G_{n,p}$ is of order $n^{-2/3+o(1)}$. We extent this result further to Walker-Breaker games and prove that this probability is also enough for Walker to create a Hamilton cycle

rr-cross tt-intersecting families via necessary intersection points

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathscr{P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of $r$-cross $t$-intersecting families. This also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for families of possibly mixed uniformities $k_1,\ldots,k_r$.Comment: 13 page

Random perturbation of sparse graphs

In the model of randomly perturbed graphs we consider the union of a deterministic graph Gα with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in Gα ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. Gα ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p)

Fast Strategies in Waiter-Client Games on KnK_n

Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page

Interactions of Colorectal Cancer, Dietary Fats, and Polymorphisms of Arachidonate Lipoxygenase and Cyclooxygenase Genes: A Literature Review

ObjectiveGenetics and dietary factors play important roles in the development of colorectal cancer (CRC). However, the underlying mechanisms of the interactions between CRC, gene polymorphisms, and dietary fat are unclear. This review study investigated the effects of polymorphisms of arachidonate lipoxygenase (ALOX) and cyclooxygenase (COX) genes in the association between CRC and dietary fat.MethodsAll the related papers published from 2000 to 2022 were collected from different databases such as PubMed, Science Direct, Scopus, and Cochran using related keywords such as colorectal cancer, ALOX, COX, polymorphism, and dietary fat. Non-English and unrelated documents were excluded.ResultsSome single-nucleotide polymorphisms (SNPs) in the ALOX and COX genes, such as rs2228065, rs6413416, and rs4986832 in the ALOX gene, and rs689465 in the COX gene may play significant roles in the association between the risk of CRC and dietary fats. SNPs of ALOX and COX genes may influence the effects of dietary fatty acids on the risk of CRC.ConclusionSome polymorphisms of the ALOX and COX genes may have important roles in the effects of dietary fat on the risk of CRC. If future studies confirm these results, dietary recommendations for preventing colorectal cancer may be personalized based on the genotype of the ALOX and COX genes

Greenhouse gas emissions after application of landfilled paper mill sludge for land reclamation of a nonacidic mine tailings site

Large areas of mine tailings are reclaimed by applying organic amendments such as paper mill sludge (PMS). Although mining industries can use PMS freshly generated by paper mills, operational constraints on paper industries make temporary landfilling of this material an unavoidable alternative for the paper industries, creating the most prominent PMS source for mining industries. This study aimed to quantify soil greenhouse gas (GHG) emissions (N2O, CO2, and CH4) after application of landfilled PMS (LPMS; i.e., excavated from a landfill site at a paper mill) and LPMS combined with a seeding treatment of white clover (Trifolium repens L.) on nonacidic mine tailings site prior to reforestation. Soil N2O, CO2, and CH4 fluxes were measured after applications of 50 and 100 Mg dry LPMS ha−1 during two consecutive snowfree seasons on two adjacent sites; LPMS was applied once in the first season. The LPMS application increased N2O emissions (7.6 to 34.7 kg N2O-N ha−1, comprising 1.04 to 2.43% of applied N) compared with the unamended control during the first season; these emissions were negligible during the second season. The LPMS application increased CO2 emissions (~5800 to 11,400 kg CO2–C ha−1, comprising 7 to 27% of applied C) compared with the unamended control on both sites and in both seasons. Fluxes of CH4 were negligible. White clover combined with LPMS treatments did not affect soil GHG emissions. These new GHG emission factors should be integrated into life-cycle analyses to evaluate the C footprint of potential symbioses between the mining and paper industries. Future research should focus on the effect of PMS applications on soil GHG emissions from a variety of mine tailings under various management practices and climatic conditions to plan responsible and sustainable land reclamation