On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most ϵ(Δ)=MΔ26 colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most ϵ(Δ) trees, and that the...

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Veröffentlicht in:	Journal of combinatorial theory. Series B 2013-07, Vol.103 (4), p.428-467
Hauptverfasser:	Chepoi, Victor, Hagen, Mark F.
Format:	Artikel
Sprache:	eng
Schlagworte:	CAT cube complex Colouring Combinatorics Contact graph Hyperplane Isometric embedding Mathematics Median graph
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container_end_page	467
container_issue	4
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container_title	Journal of combinatorial theory. Series B
container_volume	103
creator	Chepoi, Victor Hagen, Mark F.
description	We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most ϵ(Δ)=MΔ26 colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most ϵ(Δ) trees, and that the event structure whose domain is X admits a nice labelling with ϵ(Δ) labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.
doi_str_mv	10.1016/j.jctb.2013.04.003
format	Article
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source	Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals
subjects	CAT cube complex Colouring Combinatorics Contact graph Hyperplane Isometric embedding Mathematics Median graph
title	On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes
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