Abstract
Tropical curves in {mathbb {R}}^2 correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most five.
Highlights
A tropical plane curve is a metric graph, G, embedded in R2, which is dual to a regular subdivision Δ of some lattice polygon P
In the smooth case Δ is a unimodular triangulation of P, and the planar graph G is trivalent of genus g, where g agrees with the number of interior lattice points of P
Our results extend to a class of planar graphs, which is slightly more general than tropical plane curves
Summary
A tropical plane curve is a metric graph, G, embedded in R2, which is dual to a regular subdivision Δ of some lattice polygon P. For any fixed g ≥ 1, Castryck and Voight gave a procedure for finding finitely many lattice polygons with exactly g interior lattice points whose triangulations suffice to produce all such graphs (Castryck and Voight 2009) This was employed in Brodsky et al (2015) and Coles et al (2019) to computationally determine the tropically planar graphs of genus g ≤ 7. While this work is motivated by the desire to understand tropical curves and their moduli, here we concentrate on the combinatorial aspects, which means we do not take edge lengths of metric graphs into account. This comes with the advantage that our results hold for triangulations which are not regular
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