On the minimum number of bond-edge types and tile types: An approach by edge-colorings of graphs

Abstract The purpose of this paper is twofold. On one hand, we want to describe from a new graph theory perspective the self-assembly of DNA structures with branched junction molecules having flexible arms. On the other hand, we employ edge-colorings and graph decompositions to study the well-known problem of determining the minimum number of bond-edge types and tile types, which are graph invariants appearing in this context. We provide a strategy that can be applied to arbitrary graphs for obtaining upper bounds for these graph invariants.

[1]  P. Rothemund.  Folding DNA to create nanoscale shapes and patterns , 2006, Nature.

[2]  Luvena L. Ong,et al.  DNA Brick Crystals with Prescribed Depth , 2014, Nature chemistry.

[3]  C. Mao,et al.  Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra , 2008, Nature.

[4]  Greta Pangborn,et al.  Minimal Tile and Bond-Edge Types for Self-Assembling DNA Graphs , 2014 .

[5]  Natasa Jonoska,et al.  Flexible Versus Rigid Tile Assembly , 2006, UC.

[6]  Natasa Jonoska,et al.  On stoichiometry for the assembly of flexible tile DNA complexes , 2011, Natural Computing.

[7]  Giovanni Felici,et al.  Optimal pathway reconstruction on 3D NMR maps , 2015, Discret. Appl. Math..

[8]  Greta Pangborn,et al.  Using DNA Self-assembly Design Strategies to Motivate Graph Theory Concepts , 2011 .

[9]  Oswald Veblen,et al.  An Application of Modular Equations in Analysis Situs , 2022 .

[10]  Hao Yan,et al.  Structural DNA Nanotechnology: State of the Art and Future Perspective , 2014, Journal of the American Chemical Society.

[11]  Mariusz Meszka,et al.  Minimum Number of Palettes in Edge Colorings , 2014, Graphs Comb..

[12]  S. Louis Hakimi,et al.  A generalization of edge-coloring in graphs , 1986, J. Graph Theory.

[13]  Greta Pangborn,et al.  Tile-Based DNA Nanostructures: Mathematical Design and Problem Encoding , 2019 .

[14]  Greta Pangborn,et al.  Design formalism for DNA self-assembly of polyhedral skeletons using rigid tiles , 2018, Journal of Mathematical Chemistry.

[15]  Hao Yan,et al.  Challenges and opportunities for structural DNA nanotechnology. , 2011, Nature nanotechnology.

[16]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[17]  Luvena L. Ong,et al.  Three-Dimensional Structures Self-Assembled from DNA Bricks , 2012, Science.

[18]  N. Seeman.  Nucleic acid junctions and lattices. , 1982, Journal of theoretical biology.

[19]  Johannes B. Woehrstein,et al.  Polyhedra Self-Assembled from DNA Tripods and Characterized with 3D DNA-PAINT , 2014, Science.

[20]  Peter Adams,et al.  3,5‐Cycle decompositions , 1998 .