On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following "polynomial moment problem": for a complex polynomial $P(z)$ and distinct complex numbers $a,b$ to describe polynomials $q(z)$ orthogonal to all integer non-negative powers of $P(z)$ on the segment $[a,b].$Comment: 27 pages, 9 figure

Relative Tutte polynomials of tensor products of colored graphs

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollob\'as and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion-contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot