Research

Research Statement

My research is in combinatorics; more specifically but still broadly speaking, I like graph theory, graph colorings, hypergraphs, competitive optimization, and partially ordered sets.
At Illinois Tech and Ryerson, I have also been working on more applied combinatorial problems, including hypergraph algorithms, pursuit-evasion games, and complex networks.

I had been using bibtexbrowser (with extensive customizations) to automatically produce the following publication lists from bibtex files. Since I cannot use php on these pages, the lists have been transcribed directly. You can find the original scripts in index.php in the repo.

Publications (and accepted articles)

"Perfect matchings and Hamiltonian cycles in the preferential attachment model." With Alan Frieze, Xavier Pérez-Giménez and Paweł Prałat. Random Structures & Algorithms, Accepted. [doi] [arXiv]
"Dynamic coloring parameters for graphs with given genus." With Sarah Loeb, Thomas Mahoney and Jennifer Wise. Discrete Appl. Math., 235:129–141, 2018. [doi] [arXiv]
"Subhypergraphs in non-uniform random hypergraphs." With Megan Dewar, John Healy, Xavier Pérez-Giménez, Paweł Prałat, John Proos and Kirill Ternovsky†. Internet Math., 1–23, 2018. [doi] [arXiv]
"The saturation number of induced subposets of the Boolean lattice." With Michael Ferrara, Bill Kay, Lucas Kramer, Ryan R. Martin, Heather C. Smith and Eric Sullivan. Discrete Math., 340(10):2479–2487, 2017. [doi] [arXiv]
"The game saturation number of a graph." With James Carraher, William Kinnersley and Douglas B. West. J. Graph Theory, 85(2):481–495, 2017. [doi] [arXiv]
"The game of overprescribed cops and robbers played on graphs." With Anthony Bonato, Xavier Pérez-Giménez and Paweł Prałat. Graphs Combin., 33(4):801–815, 2017. [doi] [arXiv]
"Large subposets with small dimension." With Elyse Yeager. Order, 33(1):81–84, 2016. [doi] [arXiv]
"Coloring, sparseness, and girth." With Noga Alon, Alexandr V. Kostochka, Douglas B. West and Xuding Zhu. Israel J. Math., 214(1):315–331, 2016. [doi] [arXiv]
"The minimum number of edges in a 4-critical graph that is bipartite plus 3 edges." With Alexandr V. Kostochka. European J. Combin., 46:89–94, 2015. [doi]
"A note on list-coloring powers of graphs." With Nicholas Kosar, Sarka Petrickova and Elyse Yeager. Discrete Math., 332:10–14, 2014. [doi] [arXiv]
"New results on degree sequences of uniform hypergraphs." With Sarah Behrens, Catherine Erbes, Michael Ferrara, Stephen G. Hartke, Hannah Spinoza and Charles Tomlinson. Electron. J. Combin., 20(4):1–18 Paper 14, 2013. [web]
"A symplectic proof of a theorem of Franks." With Brian Collier, Ely Kerman, Bolor Turmunkh and Andrew Zimmer. Compos. Math., 148(6):1969–1984, 2012. [doi]

Submitted

"A Simple Characterization of Proportionally 2-choosable Graphs." With Hemanshu Kaul, Jeffrey Mudrock‡ and Michael Pelsmajer.
"Proportional Choosability: A New List Analogue of Equitable Coloring." With Hemanshu Kaul, Jeffrey Mudrock‡ and Michael Pelsmajer. [arXiv]

In preparation

"A degree-sequence generalization of the Sauer-Spencer packing theorem." With Hemanshu Kaul. In preparation.

† undergraduate student, ‡ graduate student

Coauthors

Noga Alon Sarah Behrens Anthony Bonato James Carraher Brian Collier Megan Dewar Catherine Erbes Michael Ferrara Alan Frieze Stephen G. Hartke John Healy Hemanshu Kaul Bill Kay Ely Kerman William Kinnersley Nicholas Kosar Alexandr V. Kostochka Lucas Kramer Sarah Loeb Thomas Mahoney Ryan R. Martin Jeffrey Mudrock Xavier Pérez-Giménez Michael Pelsmajer Sarka Petrickova Paweł Prałat John Proos Heather C. Smith Hannah Spinoza Eric Sullivan Kirill Ternovsky Charles Tomlinson Bolor Turmunkh Douglas B. West Jennifer Wise Elyse Yeager Xuding Zhu Andrew Zimmer

The above content is automatically generated from BibTeX files, using the bibtexbrowser library.

As an undergraduate I did some research in combinatorics, particularly Rosa-type graph labelings. I worked with the combinatorics REGS group at UIUC each summer, and in the 2010 summer I also worked on a REGS project concerning periodic points of Hamiltonian diffeomorphisms of the two-sphere. In the summer of 2014 I worked with the Rocky Mountains-Great Plains Graduate Research Workshop in Combinatorics.

I am broadly interested in most areas of mathematics. Areas that have attracted me most include combinatorics, geometric group theory, some category theory, cohomology of spaces and groups, and number theory.

Last modified 1 Sept 2018.

SageMath widget test:

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P=graphs.PetersenGraph()
H=P.coloring(hex_colors=True)
P.plot(vertex_colors=H)