Related papers: How to draw combinatorial map? From graphs and edg…
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially…
This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries.…
Vectors fields defined on surfaces constitute relevant and useful representations but are rarely used. One reason might be that comparing vector fields across two surfaces of the same genus is not trivial: it requires to transport the…
Tree-like tableaux are combinatorial objects that appear in a combinatorial understanding of the PASEP model from statistical mechanics. In this understanding, the corners of the Southeast border correspond to the locations where a particle…
We introduce orbital graphs and discuss some of their basic properties. Then we focus on their usefulness for search algorithms for permutation groups, including finding the intersection of groups and the stabilizer of sets in a group.
Every simple drawing of a graph in the plane naturally induces a rotation system, but it is easy to exhibit a rotation system that does not arise from a simple drawing in the plane. We extend this to all surfaces: for every fixed surface…
This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding…
Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain…
In this paper we present a novel framework for sequence to shape maps. These combinatorial maps realize exponentially many shapes, and have preimages which contain extended connected subgraphs of diameter n (neutral networks). We prove that…
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…
We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map…
Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation,…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
One of the most important combinatorial optimization problems is graph coloring. There are several variations of this problem involving additional constraints either on vertices or edges. They constitute models for real applications, such…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to…
In this paper we examine the descriptive potential of a combinatorial data structure known as "Generating Set" in constructing the boundary maps of a simplicial complex. By refining the approach of \cite{Dumas} in generating these maps, we…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…
A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…