Tropically planar graphs

Abstract

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus 7. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically 0% of connected trivalent planar graphs are tropically planar.

Characterization of tropical plane curves up to genus six

“…The graphs which do occur as skeleta of tropical curves are referred as troplanar or tropically planar [3]. Starting from the work in [1], there has been immense interest to find forbidden criteria to rule out classes of graphs which can not be realizable.…”

“…A unimodular triangulation of a genus 6 polytope (left), its dual graph (center), and the corresponding skeleton which has a double heavy cycle with two loops (right) that the skeleton as a trivalent graph is obtained from the dual tropical curve by deletion of degree one unbounded edges and smoothening the degree two edges, which are obtained after the said deletion and continuing this process till we obtain a trivalent graph. We refer the reader to [1,3,4] for further details about the duality and the graph theoretic operation to obtain the skeleton. In [1], computational techniques were employed to classify all troplanar graphs for lower genera g = 3, 4, and g = 5.…”

“…Their study provided a computational classification up to genus 5, and a theory-based classification up to genus four. In [3], this computational study is pushed to the case of genus seven and many new criteria are also established. However the classification still remained incomplete for genus five and six.…”

“…However the classification still remained incomplete for genus five and six. Some of the previously known forbidden patterns are sprawling [1], crowded [6], and TIE-fighter [3]. In [4], for the first time a complete classification of all troplanar graphs up to genus five was achieved which is independent of computational enumeration.…”

“…Figure 2. The eight trivalent planar graphs of genus six, which are not realizable [3], and remained unclassified up till now graphs. Also, we believe that with the updated list of criteria the classification of skeleta for higher genus can also be considerably reduced.…”