The cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the cyclic coloring complex of three different types of $k$-uniform hypergraphs. For the case of a complete $k$-uniform hypergraph, we show that the dimension of the $(n-k-1)^{st}$ homology group is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, ..., B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete $k$-uniform hypergraph $H$ and where $1 \in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of \C[x_1,...,x_n]/ \{x_{i_1}...x_{i_k} \mid i_{1}...i_{k} is a hyperedge of $H \}$. For the other two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show that the dimensions of the homology groups of their cyclic coloring complexes are given by binomial coefficients as well

Asymmetric 22-colorings of graphs

We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their edges can be $2$-colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface which induces a non-trivial color preserving automorphism of the graph

Genetic mechanisms of critical illness in COVID-19.

Host-mediated lung inflammation is present1, and drives mortality2, in the critical illness caused by coronavirus disease 2019 (COVID-19). Host genetic variants associated with critical illness may identify mechanistic targets for therapeutic development3. Here we report the results of the GenOMICC (Genetics Of Mortality In Critical Care) genome-wide association study in 2,244 critically ill patients with COVID-19 from 208 UK intensive care units. We have identified and replicated the following new genome-wide significant associations: on chromosome 12q24.13 (rs10735079, P = 1.65 × 10-8) in a gene cluster that encodes antiviral restriction enzyme activators (OAS1, OAS2 and OAS3); on chromosome 19p13.2 (rs74956615, P = 2.3 × 10-8) near the gene that encodes tyrosine kinase 2 (TYK2); on chromosome 19p13.3 (rs2109069, P = 3.98 ×  10-12) within the gene that encodes dipeptidyl peptidase 9 (DPP9); and on chromosome 21q22.1 (rs2236757, P = 4.99 × 10-8) in the interferon receptor gene IFNAR2. We identified potential targets for repurposing of licensed medications: using Mendelian randomization, we found evidence that low expression of IFNAR2, or high expression of TYK2, are associated with life-threatening disease; and transcriptome-wide association in lung tissue revealed that high expression of the monocyte-macrophage chemotactic receptor CCR2 is associated with severe COVID-19. Our results identify robust genetic signals relating to key host antiviral defence mechanisms and mediators of inflammatory organ damage in COVID-19. Both mechanisms may be amenable to targeted treatment with existing drugs. However, large-scale randomized clinical trials will be essential before any change to clinical practice

‘Botched labiaplasty’: a content analysis of online advertising for revision labiaplasty

A new development in female genital cosmetic surgery (FGCS) is the promotion of revision surgery for ‘botched labiaplasty’. This content analysis study reviews the quality of information offered on websites specifically advertising revision labiaplasty. Twelve websites were identified through online searches and were examined for the quality of their clinical information. All sites defined botched labiaplasty as unsatisfactory appearance after labiaplasty. Four gave no further details and five listed asymmetry, irregular labial edges or removal of too much or too little tissue. Four websites described primary botched labiaplasty as ‘mutilation’. Inadequacy of the primary surgeon was cited as the cause of botched labiaplasty in 11/12. Only two websites mentioned risks of surgery. Good outcomes were not defined and no website provided outcome data although guaranteed satisfaction was implied in two websites. This study highlights the existence and promotion of services for botched labiaplasty using non-specific and emotive descriptions. These findings suggest that unsatisfactory results from consumers’ perspectives are far from uncommon. The same women whose expectations have not been met by primary surgery are now being targeted for repeat surgery with online advertising capitalising on their unchanged motivations.Impact Statement What is already known on this subject? Female genital cosmetic surgery (FGCS) is mainly advertised online with labiaplasty as the most commonly performed procedure. A market for labiaplasty revision to correct ‘botched’ primary procedures is developing. Academic literature and advertising materials are inconsistent when defining indications and determinants of success for labiaplasty or revision. What the results of this study add? A content analysis of websites specifically advertising revision labiaplasty describes the emotive and nonspecific terms used online to promote revision labiaplasty. What the implications are of these findings for clinical practice and/or further research? The existence of services for botched labiaplasty suggests dissatisfaction is common. Women whose expectations have not been met by primary surgery are targeted for repeat surgery through online advertising capitalising on their potentially unchanged motivations. This study demonstrates the need for clearer outcome data for labiaplasty and highlights the need for better advertising standards for FGCS promotion