DATE: 11 November 2020

SPEAKER: Deniz Sarikaya (http://www.denizsarikaya.de/)

TITLE: A primer to topological infinite graph theory and how to force
Hamiltonicity in locally finite graphs via forbidden induced subgraphs

ABSTRACT: The study of Hamiltonian graphs, i.e. graphs having a cycle that
contains all vertices of the graph, is a central theme of finite graph theory.
For infinite graphs such a definition cannot work, since cycles are finite. We
shall debate possible concepts of Hamiltonicity for infinite graphs and
eventually follow the topological approach by Diestel and Kühn [1,2], which
allows to generalize several results about being a Hamiltonian graph to locally
finite graphs, i.e. graphs where each vertex has finite degree. An infinite
cycle of a locally finite connected graph G is defined as a homeomorphic image
of the unit circle S^1  in the Freudenthal compactification |G| of G. Now we
call G Hamiltonian if there is an infinite cycle in |G| containing all vertices
of G.  We examine how to force Hamiltonicity via forbidden induced subgraphs
and present recent extensions of results for Hamiltonicity in finite claw-free
graphs to locally finite ones.  This is joint work with Karl Heuer.

Bibliography
[1] R. Diestel and D. Kühn, On infinite cycles I, Combinatorica 24 (2004), pp. 69-89.
[2] R. Diestel and D. Kühn, On infinite cycles II, Combinatorica 24 (2004), pp. 91-116.