Tropical Graph Curves

TL;DR

This paper introduces tropical graph curves associated with three-regular graphs, demonstrating their existence, explicit construction, and connectedness via local operations inspired by polytope theory.

Contribution

It provides the first explicit construction of tropical graph curves for certain graphs and shows their connectedness through local transformations.

Findings

Existence of tropical graph curves for three-regular, three-vertex-connected planar graphs

Explicit construction method for these curves in tropical projective space

Connectedness of the set of tropical graph curves via local operations

Abstract

We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as \emph{tropical graph curves}. Roughly speaking, the tropical graph curve associated to $G$, whose genus is $g$, is an arrangement of $2g-2$ lines in tropical projective space that contains $G$ (more precisely, the topological space associated to $G$) as a deformation retract. We show the existence of tropical graph curves when the underlying graph is a three-regular, three-vertex-connected planar graph. Our method involves explicitly constructing an arrangement of lines in projective space, i.e. a graph curve whose tropicalisation yields the corresponding tropical graph curve and in this case, solves a topological version of the tropical lifting problem associated to canonically embedded graph curves. We also show that the set of tropical graph curves that we construct are connected via certain local operations. These local operations are inspired by Steinitz' theorem in polytope theory.