Spheres with few vertices

Venn Diagrams with Few Vertices

An n-Venn diagram is a collection of n finitely-intersecting simple closed curves in the plane, such that each of the 2n sets X1∩X2∩· · ·∩Xn, where each Xi is the open interior or exterior of the i-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight k is a k-region. A monotone Venn diagram with n curves has the property t...

Spanning trees with few branch vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all s ≥ 1, every connected graph on n vertices with minimum degree at least ( 1 s+3 + o(1))n contains a spanning tree having at most s branch vertices. Asymptotically, this is best possible and solves a problem of Flandrin, Kaiser, Kuz̆el, Li and Ryjác̆ek, which was originally motivated by an optimization problem i...

Triangulated Manifolds with Few Vertices: Combinatorial Manifolds

Centrally Symmetric Manifolds with Few Vertices

A centrally symmetric 2d-vertex combinatorial triangulation of the product of spheres S × Sd−2−i is constructed for all pairs of non-negative integers i and d with 0 ≤ i ≤ d − 2. For the case of i = d − 2 − i, the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order 4d. The crux of this construction is a defin...

Acyclic Coloring with Few Division Vertices

An acyclic k-coloring of a graph G is a mapping φ from the set of vertices of G to a set of k distinct colors such that no two adjacent vertices receive the same color and φ does not contain any bichromatic cycle. In this paper we prove that every planar graph with n vertices has a 1-subdivision that is acyclically 3-colorable (respectively, 4-colorable), where the number of division vertices i...