# Georgakopoulos, Agelos

## End-Devouring Rays ★

Author(s): Georgakopoulos

Problem   Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .

Keywords: end; ray

## Geodesic cycles and Tutte's Theorem ★★

Author(s): Georgakopoulos; Sprüssel

Problem   If is a -connected finite graph, is there an assignment of lengths to the edges of , such that every -geodesic cycle is peripheral?

Keywords: cycle space; geodesic cycles; peripheral cycles

## Hamiltonian cycles in powers of infinite graphs ★★

Author(s): Georgakopoulos

Conjecture
\item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.

Keywords: hamiltonian; infinite graph

## Hamiltonian cycles in line graphs of infinite graphs ★★

Author(s): Georgakopoulos

Conjecture
\item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.

Keywords: hamiltonian; infinite graph; line graphs