Farber's topological complexity conjecture for graph configuration spaces

Let GG be a connected graph, and let V3V_{\ge3} denote the set of vertices of GG of degree at least 33. Assume that V32|V_{\ge3}|\ge2 and let n2V3n\ge2|V_{\ge3}|. Farber's conjecture. The topological complexity of the ordered configuration space of nn points on GG satisfies

TC(Confn(G))=2V3.\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.

Sources & referencesView supporting material

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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