The degree threshold for covering with all the connected 33-graphs with 33 edges

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with $\delta_{i}(H)>c_i(n,F)$ has an $F$-covering. Falgas-Ravry, Markst\"om and Zhao (Combin. Probab. Comput., 2021) asymptotically determined $c_1(n,K_{4}^{(3)-})$, where $K_{4}^{(3)-}$ is obtained by deleting an edge from the complete $3$-graph on $4$ vertices. Later, Tang, Ma and Hou (arXiv, 2022) asymptotically determined $c_1(n,C_{6}^{(3)})$, where $C_{6}^{(3)}$ is the linear triangle, i.e. $C_{6}^{(3)}=([6],\{123,345,561\})$. In this paper, we determine $c_1(n,F_5)$ asymptotically, where $F_5$ is the generalized triangle, i.e. $F_5=([5],\{123,124,345\})$. We also determine the exact values of $c_1(n,F)$, where $F$ is any connected $3$-graphs with $3$ edges and $F\notin\{K_4^{(3)-}, C_{6}^{(3)}, F_5\}$.Comment: 17 pages, 10 figure

CRISPR/Cas9-mediated gene manipulation to create single-amino-acid-substituted and floxed mice with a cloning-free method.

Clustered regulatory interspaced short palindromic repeats (CRISPR)/CRISPR-associated protein 9 (Cas9) technology is a powerful tool to manipulate the genome with extraordinary simplicity and speed. To generate genetically modified animals, CRISPR/Cas9-mediated genome editing is typically accomplished by microinjection of a mixture of Cas9 DNA/mRNA and single-guide RNA (sgRNA) into zygotes. However, sgRNAs used for this approach require manipulation via molecular cloning as well as in vitro transcription. Beyond these complexities, most mutants obtained with this traditional approach are genetically mosaic, yielding several types of cells with different genetic mutations. Recently, a growing body of studies has utilized commercially available Cas9 protein together with sgRNA and a targeting construct to introduce desired mutations. Here, we report a cloning-free method to target the mouse genome by pronuclear injection of a commercial Cas9 protein:crRNA:tracrRNA:single-strand oligodeoxynucleotide (ssODN) complex into mouse zygotes. As illustration of this method, we report the successful generation of global gene-knockout, single-amino-acid-substituted, as well as floxed mice that can be used for conditional gene-targeting. These models were produced with high efficiency to generate non-mosaic mutant mice with a high germline transmission rate

Counting triangles in regular graphs

In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andr\'asfai-Erd\H{o}s-S\'os Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}<k<\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\frac{2n}{5}<k<\frac{n}{2}$. This confirms a conjecture of Cambie, de Verclos, and Kang