Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4


We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.

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Corresponding author

Correspondence to Radoslav Fulek.

Additional information

Supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no [291734] and by Austrian Science Fund (FWF): M2281-N35.

Supported by project 16-01602Y of the Czech Science Foundation (GAČR) and by Charles University project UNCE/SCI/004.

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Fulek, R., Kynčl, J. Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4. Combinatorica 39, 1267–1279 (2019).

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Mathematics Subject Classification (2010)

  • 05C10
  • 57M15