Matt DeVos

Evolutionarily distinctive species often capture more phylogenetic diversity than expected

Evolutionary distinctiveness measures of how evolutionarily isolated a species is relative to other members of its clade. Recently, distinctiveness metrics that explicitly incorporate time have been proposed for conservation prioritization. However, we found that such measures differ qualitatively in how well they capture the total amount of evolution (termed phylogenetic diversity, or PD) represented by a set of species. We used simulation and simple graph theory to explore this relationship with reference to phylogenetic tree shape. Overall, the distinctiveness measures capture more PD on more unbalanced trees and on trees with many splits near the present. The rank order of performance was robust across tree shapes, with apportioning measures performing best and node-based measures performing worst. A sample of 50 ultrametric trees from the literature showed the same patterns. Taken together, this suggests that distinctiveness metrics may be a useful addition to other measures of value for conservation prioritization of species. The simplest measure, the age of a species, performed surprisingly well, suggesting that new measures that focus on tree shape near the tips may provide a transparent alternative to more complicated full-tree approaches.

A minimum degree condition forcing complete graph immersion

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path Puv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths Puv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph Kt. For dense graphs one can say even more. If the graph has order n and has 2cn2 edges, then there is a strong immersion of the complete graph on at least c2n vertices in G in which each path Puv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd3/2, where c>0 is an absolute constant.For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of Kt (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture

Let R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V be the set of vertices, and for every v ∈ V, let k(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that Συ∈v k(υ) = 4π whenever R, may be realized as the surface of a convex polytope in R 3 . More generally, if R is made of finitely many polygons, Eulers formula is equivalent to the equation Συ∈vκ(υ) = 1πΧ(R) where Χ(R) is the Euler characteristic of R. Our main theorem shows that whenever Συ∈V:κ(υ) 0 for every vertex v, we apply our main theorem to deduce that R. is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless Ρ is a prism, antiprism, or the projective planar analogue of one of these that |V| < 3444. This resolves a recent conjecture of Higuchi.

Locally planar graphs are 5-choosable

It is proved that every graph embedded in a fixed surface with sufficiently large edge-width is 5-choosable.

Packing Triangles in Weighted Graphs

Tuza conjectured that for every graph

Finding one tight cycle

G

Forbidden minors for graphs with no first obstruction to parametric Feynman integration

the maximum size

Average Degree in Graph Powers

\nu

Immersing complete digraphs

of a set of edge-disjoint triangles and minimum size

Small separations in vertex-transitive graphs

\tau