The Harary-Hill conjecture states that for every n > 0 the complete graph on n vertices Kn, the minimum number of crossings over all its possible drawings equals H(n) := 1 4 ⌊n 2 ⌋⌊n− 1 2 ⌋⌊n− 2 2 ⌋⌊n− 3 2 ⌋ . So far, the lower bound of the conjecture could only be verified for arbitrary drawings of Kn with n ≤ 12. In recent years, progress has been made in verifying the conjecture for certain ...

Anders Björner characterized which finite graded partially ordered sets arise as the closure relation on cells of a finite regular CW complex. His characterization of these “CW posets” required each open interval (0̂, u) to have order complex homeomorphic to a sphere of dimension rk(u)− 2. Work of Danaraj and Klee showed that sufficient conditions were for the poset to be thin and shellable. The...

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S = k[x1, x2, ..., xn]; this includes the case of powers of the homogeneous maximal ideal (x1, x2, ..., xn) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in...

We prove that the simplicial complex Ωn of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of W -matroids for an arbitrary finite Coxeter group W . The paper generalises the well known results by Solomon-Tits and Björner on spherical buildings, and improves th...

The research summarized in this thesis consists essentially of two parts. In the first, we generalize a coloring theorem of Baxter about triangulations of the plane (originally used to prove combinatorially Brouwer's fixed point theorem in two dimensions) to arbitrary dimensions and to oriented simplicial and cubical pseudomanifolds. We show that in a certain sense no other generalizations may ...

Suppose a group G acts properly on a simplicial complex Γ . Let l be the number of G-invariant vertices, and p1,p2, . . . , pm be the sizes of the G-orbits having size greater than 1. Then Γ must be a subcomplex of Λ = Δl−1 ∗ ∂Δp1−1 ∗ · · · ∗ ∂Δpm−1. A result of Novik gives necessary conditions on the face numbers of Cohen–Macaulay subcomplexes of Λ. We show that these conditions are also suffi...

Let G = (V,E) be a graph. If G is a König graph or if G is a graph without 3-cycles and 5-cycles, we prove that the following conditions are equivalent: ∆G is pure shellable, R/I∆ is Cohen-Macaulay, G is an unmixed vertex decomposable graph and G is well-covered with a perfect matching of König type e1, . . . , eg without 4-cycles with two ei’s. Furthermore, we study vertex decomposable and she...

Given a finite simple undirected graph G there is simplicial complex Ind(G), called the independence complex, whose faces correspond to independent sets of G. This well-studied concept because it provides fertile ground for interactions between commutative algebra, theory and algebraic topology. In this paper, we consider generalization complex. [Formula: see text], subset vertex set r-independ...