# End-Devouring Rays

 Importance: Low ✭
 Author(s): Georgakopoulos, Agelos
 Subject: Graph Theory » Infinite Graphs
 Keywords: end ray
 Posted by: Agelos on: February 3rd, 2008
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 .

We say that a set of rays devours the end if every ray in meets some ray in . An end is countable if there is a countable set of rays devouring it.

If is a finite set of rays then it is not hard to prove (see [G]) that this problem has a positive answer:

Theorem   For every graph and every countable end of , if has a set of pairwise disjoint -rays, then it also has a set of pairwise disjoint -rays that devours . Moreover, can be chosen so that its rays have the same starting vertices as the rays in~ .

## Bibliography

*[G] A. Georgakopoulos, Infinite Hamilton Cycles in Squares of Locally Finite Graphs, Preprint.

* indicates original appearance(s) of problem.