Abstract. Consider n points X1, . . . ,Xn in R and denote their convex hull by Π. We prove a number of inclusion-exclusion identities for the system of convex hulls ΠI := conv(Xi : i ∈ I), where I ranges over all subsets of {1, . . . , n}. For instance, denoting by ck(X) the number of k-element subcollections of (X1, . . . ,Xn) whose convex hull contains a point X ∈ R, we prove that c1(X) − c2(...

We prove a discrete Gauss-Bonnet-Chern theorem ∑ g∈V K(g) = χ(G) for finite graphs G = (V,E), where V is the vertex set and E is the edge set of the graph. The dimension of the graph, the local curvature form K and the Euler characteristic are all defined graph theoretically.

The aim of this paper is to introduce finite spaces and their simplicial complexes. Next, we give an application to combinatorics in the form of a relation between the Euler characteristic and the Möbius function. We begin by giving an overview of finite topological spaces. We introduce beat points and weak homotopy equivalences. Then, we show that finite spaces are weak homotopy equivalent to ...

For a spanning tree T of a graph G, we define the total excess te(T, k) of T from k as te(T, k) := ∑ v∈V (T )max{dT (v)− k, 0}, where dT (v) is the degree of a vertex v in T . In this paper, we show the following; if G is a 3-connected graph on a surface with Euler characteristic χ < 0, then G has a spanning ⌈8−2χ 3 ⌉ -tree T with te(T, 3) ≤ −2χ − 1. We also show an application of this theorem ...

A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms Wh(∗ i=1 Gi) when the Gi are of finite homological type; Aut(∗ i=1...

We study random surfaces constructed by glueing together N/k filled k-gons along their edges, with all (N − 1)!! = (N − 1)(N − 3) · · · 3 · 1 pairings of the edges being equally likely. (We assume that lcm{2, k} divides N .) The Euler characteristic of the resulting surface is related to the number of cycles in a certain random permutation of {1,. .. , N }. Gamburd has shown that when 2 lcm{2, ...

(b) the canonical class K of X is ample and the intersection number (ci.Kn−i) is negative for every i ≥ 2, where ci is the i-th Chern class of X. and (c) there is a finite cover G of X isomorphic to a (P1)n−1-bundle over a nonsingular curve C. The Euler characteristic e(X) of X is equal to e(G) = 2n−1e(C). Put X ′ = X × P and L′ = p1L⊗ p2O(m + 1). Then we have, by Künneth formula, H(X ′, L′−1) ...

Let G be a graph embedded in a surface of genus g. It is shown that if the face-width of the embedding is at least clog(g)/loglog(g), then such an embedding is unique up to Whitney equivalence. If the face-width is at least clog(g), then every embedding of G which is not Whitney equivalent to our embedding has strictly sinaller Euler characteristic.

We give a classification of non-negative or Borel measurable, SL(d) invariant, homogeneous valuations on the space of d-dimensional convex polytopes containing the origin in their interiors. The only examples are volume, volume of the polar body, and the Euler characteristic.

In this thesis, we propose a method for the visualisation of overlapping sets and of fuzzy graph clusterings based on Euler diagrams. Euler diagrams are probably the most intuitive and most used method to depict sets in which elements can be shared. Such a powerful visualisation metaphor could be an invaluable visualisation tool, but the automatic generation of Euler diagrams still presents man...