The Topology of Scaffold Routings on Non-Spherical Mesh Wireframes

Authors Abdulmelik Mohammed, Nataša Jonoska, Masahico Saito

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Author Details

Abdulmelik Mohammed
  • Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA
Nataša Jonoska
  • Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA
Masahico Saito
  • Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA

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Abdulmelik Mohammed, Nataša Jonoska, and Masahico Saito. The Topology of Scaffold Routings on Non-Spherical Mesh Wireframes. In 26th International Conference on DNA Computing and Molecular Programming (DNA 26). Leibniz International Proceedings in Informatics (LIPIcs), Volume 174, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The routing of a DNA-origami scaffold strand is often modelled as an Eulerian circuit of an Eulerian graph in combinatorial models of DNA origami design. The knot type of the scaffold strand dictates the feasibility of an Eulerian circuit to be used as the scaffold route in the design. Motivated by the topology of scaffold routings in 3D DNA origami, we investigate the knottedness of Eulerian circuits on surface-embedded graphs. We show that certain graph embeddings, checkerboard colorable, always admit unknotted Eulerian circuits. On the other hand, we prove that if a graph admits an embedding in a torus that is not checkerboard colorable, then it can be re-embedded so that all its non-intersecting Eulerian circuits are knotted. For surfaces of genus greater than one, we present an infinite family of checkerboard-colorable graph embeddings where there exist knotted Eulerian circuits.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • DNA origami
  • Scaffold routing
  • Graphs
  • Surfaces
  • Knots
  • Eulerian circuits


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