Flexible placements of periodic graphs in the plane

Sean Dewar

Published 2019-11-13Version 1

Given a periodic graph, we wish to determine via combinatorial methods whether it has perioidic embeddings in the plane that are flexible, i.e. allow motions that preserve edge-lengths and periodicity to non-congruent embeddings. By introducing NBAC-colourings for the corresponding quotient gain graphs, we identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed. We further characterise with NBAC-colourings which 1-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity, and characterise in special cases which 2-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity.