189 research outputs found
The degree threshold for covering with all the connected -graphs with edges
Given two -uniform hypergraphs and , we say that has an
-covering if every vertex in is contained in a copy of . Let
be the least integer such that every -vertex -graph with
has an -covering. Falgas-Ravry, Markst\"om and Zhao
(Combin. Probab. Comput., 2021) asymptotically determined
, where is obtained by deleting an edge
from the complete -graph on vertices. Later, Tang, Ma and Hou (arXiv,
2022) asymptotically determined , where is
the linear triangle, i.e. . In this paper,
we determine asymptotically, where is the generalized
triangle, i.e. . We also determine the exact values
of , where is any connected -graphs with edges and
.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
, in -vertex -regular graphs, where is an odd integer and
is an even integer. The well-known Andr\'asfai-Erd\H{o}s-S\'os Theorem has
established that if . In a striking work, Lo has
provided the exact value of for sufficiently large , given that
. Here, we bridge the gap
between the aforementioned results by determining the precise value of
in the entire range . This confirms a conjecture of
Cambie, de Verclos, and Kang
- β¦