Speciation is characterized by the development of reproductive isolating barriers between diverging groups. Intrinsic post-zygotic barriers of the type envisioned by Bateson, Dobzhansky, and Muller are deleterious epistatic interactions among loci that reduce hybrid fitness, leading to reproductive isolation. The first formal population genetic model of the development of these barriers was published by Orr in 1995, and here we develop a more general model of this process by incorporating finite protein–protein interaction networks, which reduce the probability of deleterious interactions in vivo. Our model shows that the development of deleterious interactions is limited by the density of the protein–protein interaction network. We have confirmed our analytical predictions of the number of possible interactions given the number of allele substitutions by using simulations on the Saccharomyces cerevisiae protein–protein interaction network. These results allow us to define the rate at which deleterious interactions are expected to form, and hence the speciation rate, for any protein–protein interaction network.

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The crossing number normalCR ( G ) of a graph G goodbreakinfix= ( V , E ) is the smallest number of edge crossings over all drawings of G in the plane. For any k goodbreakinfix≥ 1, the k‐planar crossing number of G goodbreakinfix, CR k ( G ), is defined as the minimum of CR ( G 1 ) + CR ( G 2 ) + ⋯ + CR ( G k ) over all graphs G 1 goodbreakinfix, G 2 , … , G k with ∪ i = 1 k G i goodbreakinfix= G. Pach et al [Comput. Geom.: Theory Appl. 68 (2018), pp. 2–6] showed that for every k goodbreakinfix≥ 1, we have CR k ( G ) ≤ ( 2 / k 2 − 1 / k 3 ) CR ( G ) and that this bound does not remain true if we replace the constant 2 / k 2 − 1 / k 3 by any number smaller than 1 / k 2. We improve the upper bound to ( 1 / k 2 ) ( 1 + o ( 1 ) ) as k goodbreakinfix→ ∞. For the class of bipartite graphs, we show that the best constant is exactly 1 / k 2 for every k. The results extend to the rectilinear variant of the k‐planar crossing number.

It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745-756] that for n ≥ 5 every simple graph of order n and size at least n 2 − n + 5 has an orientation of diameter two. We prove this conjecture and hence determine for every n ≥ 5 the minimum value of m such that every graph of order n and size m has an orientation of diameter two.diameter and oriented diameter and orientation and oriented graph and distance and size