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End-Devouring Rays

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

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

If $ K $ 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 $ G $ and every countable end $ \omega $ of $ G $, if $ G $ has a set $ K $ of $ k\in \mathcal N $ pairwise disjoint $ \omega $-rays, then it also has a set $ K' $ of $ k $ pairwise disjoint $ \omega $-rays that devours $ \omega $. Moreover, $ K' $ can be chosen so that its rays have the same starting vertices as the rays in~$ K $.

Bibliography

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


* indicates original appearance(s) of problem.
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