Minimal Translation Covers For Sets of Diameter 1

In this note we prove that the smallest perimeter convex domain that is a translation cover for the collection of all planar sets of diameter 1 is the circle of diameter .     

Noncommutative schemes

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: ((X, O), Spec(A)) Hom (A, (X, O)). We define a noncommutative version of the ech cohomology of an affine cover and show that the ech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.     

Projective covers and minimal sets of generators over local rings

We describe the structure of projective covers of modules over a local ring, when such covers exist, and modules with minimal sets of generators. The case of modules over valuation rings is studied in more detail.     

On the powers of vertex cover ideals

Let $S=\mathbb{K}[x_1,\dots ,x_n]$ be a polynomial ring, where $\mathbb{K}$ is a field, and G be a simple graph on n vertices. Let $J(G)?S$ be the vertex cover ideal of G. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph, then symbolic powers of $J(G)$ are componentwise linear.     

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function $E(G,x)={\sum }_{i\ge 1}e(G,i)x^i$, where $e(G,i)$ is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph $K_n$, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least $n?3$ non-real roots.     

Covering groups of nonconnected topological groups and 2-groups

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are equal.     

Another two graphs with no planar covers

A graph H is a cover of a graph G if there exists a mapping φ from V( H ) onto V( G ) such that φ maps the neighbors of every vertex υ in H bijectively to the neighbors of φ(υ) in G . Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the results of Archdeacon, Fellows, Negami, and the author that the conjecture holds as long as K _1,2,2,2 has no finite planar cover. However, this is still an open question, and K _1,2,2,2 is not the only minor‐minimal graph in doubt. Let ??₄ (?₂) denote the graph obtained from K _1,2,2,2 by replacing two vertex‐disjoint triangles (four edge‐disjoint triangles) not incident with the vertex of degree 6 with cubic vertices. We prove that the graphs ??₄ and ?₂ have no planar covers. This fact is used in [P. Hlin?ný, R. Thomas, On possible counterexamples to Negami's planar cover conjecture, 1999 (submitted)] to show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 227–242, 2001     

General properties of some families of graphs defined by systems of equations

In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and biregularity, existence of special vertex colorings, and existence of covering maps—hence, embedded spectra—between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge‐decomposes either the complete, or complete bipartite, graph which it spans. In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edge‐decomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 65–86, 2001     

Stability of arc‐transitive graphs

The present paper investigates arc‐transtive graphs in terms of their stability, and shows, somewhat contrary to expectations, that the property of instability is not as rare as previously thought. Until quite recently, the only known example of a finite, arc‐transitive vertex‐determining unstable graph was the underlying graph of the dodecahedron. This paper illustrates some methods for constructing finite arc‐transitive unstable graphs, and three infinite families of such graphs are given as applications. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 95–110, 2001