Farber's topological complexity conjecture for graph configuration spaces

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Let GG be a connected graph, and let V≥3V_{\ge3} denote the set of vertices of GG of degree at least 33. Assume that ∣V≥3∣≥2|V_{\ge3}|\ge2 and let n≥2∣V≥3∣n\ge2|V_{\ge3}|. Farber's conjecture. The topological complexity of the ordered configuration space of nn points on GG satisfies

TC⁡(Conf⁡n(G))=2∣V≥3∣.\operatorname{TC}(\operatorname{Conf}_n(G))=2|V_{\ge3}|.

This conjecture predicts that, once the number of particles is sufficiently large, topological complexity is determined by the number of essential vertices of the graph. The supplied source does not state whether the conjecture has been resolved.

References

Primary source

Daniel Lütgehetmann and David Recio-Mitter, “Topological complexity of configuration spaces of fully articulated graphs and banana graphs”, arXiv:1806.00659 (2019).

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