Tropically planar graphs number 152 at genus 6 and 672 at genus 7, grow between roughly 2.47^g and 2^{11g/3+o(g)}, and make up asymptotically 0% of connected trivalent planar graphs.
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Tropically planar graphs
Tropically planar graphs number 152 at genus 6 and 672 at genus 7, grow between roughly 2.47^g and 2^{11g/3+o(g)}, and make up asymptotically 0% of connected trivalent planar graphs.