The Genus of Curve, Pants and Flip Graphs

Abstract

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface. The lower bounds are shown using probabilistic methods.

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Correspondence to Bram Petri.

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Research partially supported by Swiss National Science Foundation grant number PP00P2_153024.

Editor in Charge: János Pach

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Parlier, H., Petri, B. The Genus of Curve, Pants and Flip Graphs. Discrete Comput Geom 59, 1–30 (2018). https://doi.org/10.1007/s00454-017-9922-7

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Keywords

  • Curve graph
  • Pants graph
  • Flip graph
  • Graph genus
  • Surfaces

Mathematics Subject Classification

  • Primary 57M15
  • Secondary 05C10
  • 05C80