Topological Graph Theory from Japan

Topological Graph Theory A Survey

Some Topological Methods in Graph Coloring Theory

Attempts to solve the famous Four Color Problem led to fruitful discoveries and rich coloring theories. In this talk, some old and some recent applications of tools from topology to graph coloring problems will be presented. In particular, the following subjects will be treated: The use of Euler’s formula and local planarity conditions, Kempe equivalence, homotopy, winding number and its higher...

Recent Results in Topological Graph Theory*

A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, torus, projective plane etc. The purpose of this lecture is to explore some of these topological a...

A Fundamentally Topological Perspective on Graph Theory

We adopt a novel topological approach for graphs, in which edges are modelled as points as opposed to arcs. The model of “classical” topologized graphs translates graph isomorphism into topological homeomorphism, so that all combinatorial concepts are expressible in purely topological language. This allows us to extrapolate concepts from finite graphs to infinite graphs equipped with a compatib...

Topological graph polynomials and quantum field theory

We define a new topological polynomial extending the Bollobás-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theor...