Coalescence, genetic diversity in sexual populations under selection

In sexual populations, selection operates neither on the whole genome, which is repeatedly taken apart and reassembled by recombination, nor on individual alleles that are tightly linked to the chromosomal neighborhood. The resulting interference between linked alleles reduces the efficiency of selection and distorts patterns of genetic diversity. Inference of evolutionary history from diversity shaped by linked selection requires an understanding of these patterns. Here, we present a simple but powerful scaling analysis identifying the unit of selection as the genomic "linkage block" with a characteristic length determined in a self-consistent manner by the condition that the rate of recombination within the block is comparable to the fitness differences between different alleles of the block. We find that an asexual model with the strength of selection tuned to that of the linkage block provides an excellent description of genetic diversity and the site frequency spectra when compared to computer simulations. This linkage block approximation is accurate for the entire spectrum of strength of selection and is particularly powerful in scenarios with many weakly selected loci. The latter limit allows us to characterize coalescence, genetic diversity, and the speed of adaptation in the infinitesimal model of quantitative genetics

K_6 minors in 6-connected graphs of bounded tree-width

We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6-connected graphs. Jorgensen conjectured that it holds for all 6-connected graphs.Comment: 33 pages, 8 figure

K6minors in 6-connected graphs of bounded tree-width

We prove that every sufficiently large 6-connected graph of bounded tree-width either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6-connected graphs. Jørgensen conjectured that it holds for all 6-connected graphs