# Category: Topological graph theory

Petrie dual
In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces. The
Xuong tree
In graph theory, a Xuong tree is a spanning tree of a given graph with the property that, in the remaining graph , the number of connected components with an odd number of edges is as small as possibl
Dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separ
Graph manifold
In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhe
Topological graph theory
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies im
Book embedding
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. Usually, the ver
Queue number
In the mathematical field of graph theory, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of
Crossing Numbers of Graphs
Crossing Numbers of Graphs is a book in mathematics, on the minimum number of edge crossings needed in graph drawings. It was written by Marcus Schaefer, a professor of computer science at DePaul Univ
Topological graph
In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jo
Matsushima's formula
In mathematics, Matsushima's formula, introduced by Matsushima, is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of th
Turán's brick factory problem
Unsolved problem in mathematics: Can any complete bipartite graph be drawn with fewer crossings than the number given by Zarankiewicz? (more unsolved problems in mathematics) In the mathematics of gra
Graph-encoded map
In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices per edge of the original graph. It is the top
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles
Heawood conjecture
In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. For surfaces of genus
In graph theory the road coloring theorem, known previously as the road coloring conjecture, deals with synchronized instructions. The issue involves whether by using such instructions, one can reach
Three utilities problem
The classical mathematical puzzle known as the three utilities problem or sometimes water, gas and electricity asks for non-crossing connections to be drawn between three houses and three utility comp
Left-right planarity test
In graph theory, a branch of mathematics, the left-right planarity testor de Fraysseix–Rosenstiehl planarity criterion is a characterization of planar graphs based on the properties of the depth-first
Rotation system
In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a num
Cycle double cover
In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Euler's Gem
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written b
String graph
In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if and only if there exists a set of curves, or s
Albertson conjecture
In combinatorial mathematics, the Albertson conjecture is an unproven relationship between the crossing number and the chromatic number of a graph. It is named after Michael O. Albertson, a professor
Heawood number
In mathematics, the Heawood number of a surface is an upper bound for the number of colors that suffice to color any graph embedded in the surface. In 1890 Heawood proved for all surfaces except the s
Dessin d'enfant
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational number
Toroidal graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
Ribbon graph
In topological graph theory, a ribbon graph is a way to represent graph embeddings, equivalent in power to signed rotation systems or graph-encoded maps. It is convenient for visualizations of embeddi
Covering space
A covering of a topological space is a continuous map with special properties.
Combinatorial map
A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon gra
Sequence covering map
In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the
Thrackle
A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arcand every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoi
Graph embedding
In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs (homeomorphic image
Antiprism
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. The
Generalized map
In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent n
Regular map (graph theory)
In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projecti
Crossing number inequality
In the mathematics of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the minimum number of crossings of a given graph, as a function of the number of edges and
Crossing number (graph theory)
In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is
Wilson operation
In topological graph theory, the Wilson operations are a group of six transformations on graph embeddings. They are generated by two involutions on embeddings, surface duality and Petrie duality, and