Page 250 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 250
APPLIED COMBINATORIAL AND GEOMETRIC TOPOLOGY (MS-34)
Hamiltonian complexes, interval graphs, determinantal ideals
Bruno Bendetti, bruno@math.miami.edu
University of Miami, United States
Coauthors: Lisa Seccia, Matteo Varbaro
There are three mutually-related notions in graph theory, from three different centuries: Hamil-
tonian paths, (unit)-interval graphs, and binomial edge ideals. We study the d-dimensional
generalization of these ideas.
This is joint work with Matteo Varbaro and Lisa Seccia (arXiv:2101.09243).
A colored approach for the self-assembly of DNA structures
Simona Bonvicini, sbonvicini@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Margherita Maria Ferrari
We study a graph theory problem related to the self-assembly of DNA structures. The self-
assembly can be obtained by several methods that are based on the Watson-Crick complemen-
tary properties of DNA strands. We consider the method based on branched junction molecules,
that is, star-shaped molecules whose arms have cohesive ends that allow the molecules to join
together in a prescribed way and form a larger molecule (DNA complex).
In the language of graphs, a branched junction molecule is called a tile and consists of a
vertex with labeled half-edges; labels represent the cohesive ends and belong to a set {a, aˆ :
a ∈ Σ}, where Σ is a finite set of symbols; a tile is denoted by the multiset consisting of the
labels of the half-edges; and two tiles are of the same tile type if they are denoted by the same
multiset.
We can create an edge between the vertices u, v if and only if u has a half-edge labeled by
a and v has a half-edge labeled by aˆ; the edge thus obtained is said to be a bond-edge of type
aaˆ; by connecting the vertices according to the labels, we can construct a graph G representing
a DNA complex.
The following problem is considered: given a graph G, determine the minimum number of
tile types and bond-edge types that are necessary to construct G. We describe the problem by
edge-colored graphs and show some upper bounds for the number of bond-edge types that are
necessary to construct an arbitrary graph.
Universality classes of triangulations in dimensions greater than 2
Valentin Bonzom, bonzom@lipn.univ-paris13.fr
LIPN, CNRS UMR 7030, Institut Galilée, Université Sorbonne Paris Nord, France
Combinatorial maps are cellular topological models for surfaces, which include triangulations,
quadrangulations, etc. of all genera. It is well-known that at fixed topology, all models lie in
the same universality class. It means in particular that they have the same critical exponents in
the enumeration formula and the same limit object at large scales. For instance, the universality
class for planar maps is also known as the universality class of 2D pure quantum gravity and its
scaling limit is known as the Brownian sphere. It is important to generalize this framework to
248
Hamiltonian complexes, interval graphs, determinantal ideals
Bruno Bendetti, bruno@math.miami.edu
University of Miami, United States
Coauthors: Lisa Seccia, Matteo Varbaro
There are three mutually-related notions in graph theory, from three different centuries: Hamil-
tonian paths, (unit)-interval graphs, and binomial edge ideals. We study the d-dimensional
generalization of these ideas.
This is joint work with Matteo Varbaro and Lisa Seccia (arXiv:2101.09243).
A colored approach for the self-assembly of DNA structures
Simona Bonvicini, sbonvicini@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthor: Margherita Maria Ferrari
We study a graph theory problem related to the self-assembly of DNA structures. The self-
assembly can be obtained by several methods that are based on the Watson-Crick complemen-
tary properties of DNA strands. We consider the method based on branched junction molecules,
that is, star-shaped molecules whose arms have cohesive ends that allow the molecules to join
together in a prescribed way and form a larger molecule (DNA complex).
In the language of graphs, a branched junction molecule is called a tile and consists of a
vertex with labeled half-edges; labels represent the cohesive ends and belong to a set {a, aˆ :
a ∈ Σ}, where Σ is a finite set of symbols; a tile is denoted by the multiset consisting of the
labels of the half-edges; and two tiles are of the same tile type if they are denoted by the same
multiset.
We can create an edge between the vertices u, v if and only if u has a half-edge labeled by
a and v has a half-edge labeled by aˆ; the edge thus obtained is said to be a bond-edge of type
aaˆ; by connecting the vertices according to the labels, we can construct a graph G representing
a DNA complex.
The following problem is considered: given a graph G, determine the minimum number of
tile types and bond-edge types that are necessary to construct G. We describe the problem by
edge-colored graphs and show some upper bounds for the number of bond-edge types that are
necessary to construct an arbitrary graph.
Universality classes of triangulations in dimensions greater than 2
Valentin Bonzom, bonzom@lipn.univ-paris13.fr
LIPN, CNRS UMR 7030, Institut Galilée, Université Sorbonne Paris Nord, France
Combinatorial maps are cellular topological models for surfaces, which include triangulations,
quadrangulations, etc. of all genera. It is well-known that at fixed topology, all models lie in
the same universality class. It means in particular that they have the same critical exponents in
the enumeration formula and the same limit object at large scales. For instance, the universality
class for planar maps is also known as the universality class of 2D pure quantum gravity and its
scaling limit is known as the Brownian sphere. It is important to generalize this framework to
248
