Related papers: Forbidden Patterns in Tropical Plane Curves
We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain…
We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of…
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…
In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…
Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case…
We characterize permutations whose Bruhat graphs can be drawn in the plane and those whose Bruhat graphs can be drawn in the torus. In particular, we show these properties are characterized by avoiding finitely many permutations.
We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts…
In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…
We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erd\H{o}s. We also mention some related problems.
A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can…
A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves…
We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the…
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we…
We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
An interval graph is considered improper if and only if it has a representation such that an interval contains another interval. Previously these have been investigated in terms of balance and minimal forbidden interval subgraphs for the…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…