Related papers: What can you draw?

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial…

In this paper we define the combinatorial analogous of a pencil, and show its relationship with the concept of admissibility. Such an object is usefull to study the isomorphisms between fundamental groups of the complements of line…

Cutting planes are a key ingredient to successfully solve mixed-integer linear programs. For specific problems, their strength is often theoretically assessed by showing that they are facet-defining for the corresponding mixed-integer hull.…

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings…

In the paper we characterize subspaces and pencils of lines of generalized (and multidimensional) Laguerre spaces and we consider definability of the structure of "conic" pencils in the Grassmann space of 1-subspaces. We also study…

Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic…

A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): Is it true that if the total width of a collection of planks…

A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…

Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree at least 13.

Holographic techniques significantly extend the capabilities of laser tweezing, making possible extended trapping patterns for manipulating large numbers of particles and volumes of soft matter. We describe practical methods for creating…

We prove that every set $\mathcal S$ of $\Delta$ slopes containing the horizontal slope is universal for $1$-bend upward planar drawings of bitonic $st$-graphs with maximum vertex degree $\Delta$, i.e., every such digraph admits a $1$-bend…

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…

Kolmogorov asked the following question: can every bounded measurable set in the plane be mapped onto a polygon by a 1-Lipschitz map with arbitrarily small measure loss? The answer is negative in general, however, the case of compact sets…

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly…

We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…

This paper investigates the possibility of constructive extraction of measurable selector from set-valued maps which may commonly arise in viability theory, optimal control, discontinuous systems etc. For instance, existence of solutions to…

Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question if a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is…

A linear subspace $E$ of $\mathbb{R}^n$ has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are weak if the digitizations can slightly wander around $E$. We prove…

We prove a conjecture of Dvo\v{r}\'ak, Kr\'al, Nejedl\'y, and \v{S}krekovski that planar graphs of girth at least five are square $(\Delta+2)$-colorable for large enough $\Delta$. In fact, we prove the stronger statement that such graphs…