# Georgakopoulos, Agelos

## End-Devouring Rays ★

Author(s): Georgakopoulos

\begin{problem} Let $G$ be a graph, $\omega$ a countable end of $G$, and $K$ an infinite set of pairwise disjoint $\omega$-rays in $G$. Prove that there is a set $K'$ of pairwise disjoint $\omega$-rays that devours $\omega$ such that the set of starting vertices of rays in $K'$ equals the set of starting vertices of rays in $K$. \end{problem}

## Geodesic cycles and Tutte's Theorem ★★

Author(s): Georgakopoulos; Sprüssel

\begin{problem} If $G$ is a $3$-connected finite graph, is there an assignment of lengths $\ell: E(G) \to \mathb R^+$ to the edges of $G$, such that every $\ell$-geodesic cycle is \Def[peripheral]{peripheral cycle}? \end{problem}

Keywords: cycle space; geodesic cycles; peripheral cycles

## Hamiltonian cycles in powers of infinite graphs ★★

Author(s): Georgakopoulos

\begin{conjecture} \begin{enumerate} \item If $G$ is a countable connected graph then its third \Def[power]{Glossary_of_graph_theory#Distance} is hamiltonian. \item If $G$ is a 2-connected countable graph then its square is hamiltonian. \end{enumerate} \end{conjecture}

Keywords: hamiltonian; infinite graph

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

Author(s): Georgakopoulos

\begin{conjecture} \begin{enumerate} \item If $G$ is a 4-edge-connected locally finite graph, then its \Def{line graph} is hamiltonian. \item If the line graph $L(G)$ of a locally finite graph $G$ is 4-connected, then $L(G)$ is hamiltonian. \end{enumerate} \end{conjecture}

Keywords: hamiltonian; infinite graph; line graphs