Cycles through specified vertices in triangle-free graphs

Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V(G)|+2. Let S ⊂ V(G) consist of less than σ4/4+ 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ4 are best possible

Fine-scale biogeographical and temporal diversification processes of peacock swallowtails (<em>Papilio</em> subgenus <em>Achillides</em>) in the Indo-Australian Archipelago

International audienceExplanations for the high species diversity of the Indo-Australian Archipelago are often challenged by the regions complex climatic and geological histories. Here, we investigated the evolutionary history of swallowtail butterflies of the Papilio subgenus Achillides, comprising up to 25 recognized species and about 100 subspecies distributed across the Indo-Australian Archipelago. To estimate the relative contributions of factors influencing their biodiversity, we used DNA sequences to infer the phylogeny and species limits of 22 species including most of their subspecies. We recovered a highly resolved and well-supported phylogeny for the subgenus, and clarified some taxonomic ambiguities at the species level. The corresponding DNA-based species phylogeny was then employed to reconstruct their historical biogeography using relaxed-clock and parametric-based analyses. Molecular dating and biogeographical analyses showed that Achillides originated around 19 Ma in Sunda + Wallacea. Biogeographical reconstructions indicated that geological vicariance shaped the early evolutionary history of Achillides whereas dispersal influenced late diversification. Birthdeath likelihood analyses allowed exploration of their tempo and mode of diversification. We detected several shifts in diversification rates that are attributable to past climate-induced biogeographical events. By assessing both regional and fine-scale biodiversity patterns, this study brings new findings to a biogeographical understanding of the Indo-Australian Archipelago