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On dihedral flows in embedded graphs.

Bart Litjens1

  • 1Korteweg-de Vries Institute for Mathematics University of Amsterdam Amsterdam Netherlands.

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PubMed
Summary
This summary is machine-generated.

This study investigates graph flows, specifically nowhere-identity dihedral flows. Counterexamples were found, challenging previous conjectures and revealing obstructions for cubic graphs.

Keywords:
cubic graphdihedral groupembedded graphflownonabelian flow

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Area of Science:

  • Graph theory
  • Combinatorics
  • Algebraic topology

Background:

  • The study of graph flows, particularly nowhere-identity flows, is crucial in understanding graph structures.
  • The connection between nowhere-identity dihedral flows and nowhere-identity flows in multigraphs has been a recent area of inquiry.

Purpose of the Study:

  • To investigate the equivalence between the existence of nowhere-identity dihedral flows and nowhere-identity flows in multigraphs.
  • To provide counterexamples and general obstructions to a conjecture by Goodall et al. (2016).
  • To analyze the computational complexity of deciding the existence of nowhere-identity flows.

Main Methods:

  • Construction of counterexamples for the conjecture.
  • Development of general obstruction techniques for nowhere-identity flows.
  • Complexity analysis using theoretical computer science methods.
  • Focus on specific graph classes, particularly cubic graphs.

Main Results:

  • Counterexamples demonstrating that a multigraph admits a nowhere-identity dihedral flow if and only if it admits a nowhere-identity flow with is not universally true.
  • Identification of general obstructions that prevent the existence of such flows.
  • Characterization of graphs for which the equivalence holds.
  • Discussion of the complexity of determining the existence of nowhere-identity flows.

Conclusions:

  • The conjecture by Goodall et al. is disproven.
  • Obstructions and specific graph properties determine the existence of nowhere-identity dihedral flows.
  • The computational complexity of related flow problems is significant.