[4] On the scramble number of graphs [arXiv:2103.15253] (to appear in Discrete Applied Mathematics)

The scramble number of a graph is an invariant recently developed to aid in the study of divisorial gonality. In this paper we prove that scramble number is NP-hard to compute, also providing a proof that computing gonality is NP-hard even for simple graphs, as well as for metric graphs. We also provide general lower bounds the scramble number of a Cartesian product of graphs, and apply these to compute gonality for many new families of product graphs.

[3] The Rest of the Tilings of the Sphere by Regular Polygons [arXiv:2101.10743] (submitted)

We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.

[2] Moduli dimensions of lattice polygons [arXiv:2010.13135] (published in Journal of Algebraic Combinatorics)

Given a lattice polygon P with g interior lattice points, we associate to it the moduli space of tropical curves of genus g with Newton polygon P. We completely classify the possible dimensions such a moduli space can have. For non-hyperelliptic polygons the dimension must be between g and 2g+1, and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to 2g−1. In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P.

[1] Prism graphs in tropical plane curves [arXiv:2009.08570] (published in Involve Journal of Mathematics)

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus 12 and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most 11. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others.

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