Recent results in topological graph theory

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...

Recent results on the majorization theory of graph spectrum and topological index theory

Suppose π = (d1, d2, . . . , dn) and π = (d1, d ′ 2 , . . . , dn) are two positive nonincreasing degree sequences, write π ⊳ π if and only if π 6= π, ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 di for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G be the extremal graphs with the maximal (signless...

Recent research in Graph Theory

A well-known and fundamental property of graphs is Hamiltonicity. A connected graph is Hamiltonian if it contains a spanning cycle. Determining if a graph is Hamiltonian is known as a NP-complete problem and no satisfactory characterization exists. Nevertheless, many sufficient conditions for Hamiltonicity were found, usually expressed in terms of degree, connectivity, density, toughness, indep...

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...