Combinatorics, Graph Theory, and More

My interests lie primarily within combinatorics, the area of mathematics which studies finite discrete structures. Recently, my research has focused on new aspects of the forbidden subgraph problem. However, I am also very interested in the use of probability in deterministic settings, as well as a wide variety of probabilistic and extremal questions on sets, graphs, and hypergraphs.

Have math, will travel.

I love speaking about math, to audiences at all levels. If you're looking for a seminar, colloquium, math circle, outreach activity, or something else, send me an e-mail (nobushaw (at) vcu (dot) edu). I've recently put together a talk about "Combinatorial Music Theory" designed for a general mathematical audience, so let me know if that's something you're interested in.

Upcoming Talks / Conferences:

  • Oct 25, 2025: "Bootstrap Percolation on Grids and Tori", AMS Special Session on Recent Trends in Graph Theory (Virtual).
  • Nov 1, 2025: "I heard there was a secret chord...", Shenandoah Undergraduate Mathematics and Statistics Conference, James Madison University, Harrisonberg VA (Plenary).
  • YOUR SEMINAR?!?!

Session Organization:

Papers & Other Writing

  • N. Bushaw, P. Csorba, L. Erickson, D. Gerbner, D. Piguet, A. Riet, T. Terpai, D.K. Vu."Large Matchings with Few Colors". Technical Report. [arXiv]
  • N. Bushaw, N. Kettle. "Turan numbers of Multiple Paths and Equibipartite Trees." Combinatorics, Probability and Computing, 20 (2011) pp 837-853. [arXiv]
  • N. Bushaw. "Problems in Extremal Combinatorics." PhD Dissertation. Major Professor: Dr. Bela Bollobas, University of Memphis. May 2012. [pdf]
  • N. Bushaw, N. Kettle. "Turan numbers for Forests of Paths in Hypergraphs". SIAM J. Discrete Math., 28-2 (2014) pp 711-721. [arXiv]
  • N. Bushaw, M. Collares Neto, R. Morris, P. Smith. "The sharp threshold for maximum-size sum-free subsets in even-order abelian groups". Combinatorics, Probability, and Computing 20, no. 4 (2015) pp 609-640. [arXiv]
  • N. Bushaw, K. Gunderson, S. Kalikow. "Random-step Markov Processes". Israel Journal of Mathematics 216 (1) (2016) pp 181-214. [arXiv]
  • J. Balogh, N. Bushaw, M. Collares Neto, H. Liu, R. Morris, M. Sharifzadeh. "The Typical Structure of Graphs with No Large Cliques". Combinatorica 37 (4) (2017) pp 617-632.[arXiv]
  • N. Bushaw, D. Cranston. "A Note on Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons". Australasian Journal of Combinatorics, Vol 74 (3), pp. 486--497 (2019). [arXiv] [PDF]
  • N. Bushaw, N. Kettle. "Turan numbers for Forestable Graphs". Graphs and Combinatorics (36), pp. 459--467 (2020). [Journal Link]
  • N. Bushaw, B. Conka, V. Gupta, A. Kierans, H. Lafayette, C. Larson, K. McCall, A. Mulyar, C. Sullivan, S. Taylor, E. Wainright, E. Wilson, G. Wu. "Bootstrap Percolation via Automated Conjecturing". Ars Mathematica Contemporanea, [S.I] (2022)
  • N. Bushaw, D. Johnston, P. Rombach. "Rainbow Saturation". Graphs and Combinatorics (38-5), pp. 1--12 (2020). [arXiv]
  • N. Bushaw, A. Czygrinow, J. Yie. "Even Cycles in Dense Graphs", Technical Report, [arXiv]
  • N. Bushaw, V. Gupta, C. Larson, S. Loeb, M. Norge, J. Parrish, J. Yirka, G. Wu. "New conditions for Graph Hamiltonicity. (To appear: Involve)
  • N. Bushaw, V. Bednar. Rainbow Turán Methods for Trees. (To appear: Graphs & Combinatorics) [arXiv]
  • N. Bushaw, N. Kettle. Thresholds for Pebbling on Grids.. (To appear: Discrete Math) [arXiv]
  • N. Bushaw, C. Larson, N. Van Cleemput, and Summer 2017 Graph Brain Project Workshop Participants. Automated Conjecturing VII: The Graph Brain Project & Big Mathematics. (Submitted)
  • N. Bushaw, G. Hurlbert. Thresholds for zero sums with small cross number in abelian groups. (To appear: Integers) [arXiv]
  • N. Bushaw, B. Cody, L. Freeman, T. Whitaker. The Music and Mathematics of Maximal Evenness in Graphs. (To appear: Bridges Conference proceedings). [PDF]
  • N. Bushaw, B. Cody, L. Freeman. Maximal Evenness in Graphs. (To appear: Discrete Math) [arXiv]
  • N. Bushaw, S. English, E. Heath, D. Johnston, P. Rombach. The Saturation Spectrum of Berge Stars. (Submitted) [arXiv]
  • V. Bardenova, N. Bushaw, B. Cody, P. Fay, M. Tennant. The Wiener index of vertex colorings. (Submitted) [arXiv]
  • N. Bushaw, J. Danielsson, G. Hurlbert. Erdos-Ko-Rado Theorems for Paths in Graphs. (Submitted) [arXiv]

Mathematical Art

  • Alternative Pianos, , Bridges 2024 Exhibition of Mathematical Art, Craft, and Design, Richmond VA (2024).

    • Past Research Talks:

    (In Aug 2022, this was declared a "highlights" section, and is no longer guaranteed to be exhaustive. There are talks which are lost to the sands of time -- if you have a hint of a memory of a vision of seeing me give a talk somewhere and it isn't in this list, it is best to assume it probably happened. Or you could invite me for a repeat.)
    • Oct 2025. "Bootstrap Percolation on Grids and Tori". VCU Discrete Math Seminar, Richmond VA.
    • Oct 2025. "The Hypergraph Saturation Spectrum for Berge Stars". AMS Special Session on Recent Trends in Graph Theory, New Orleans, LA.
    • Apr 2025. "Maximally Even Sets", UVm-Dartmouth Discrete Math Day, Burlington VT.
    • Apr 2025. "I Heard There Was A Secret Chord", Colloquium, University of Vermont, Burlington VT.
    • Feb 2025. "Intersecing Families of Paths", VCU Discrete Math Seminar, Richmond VA.
    • Dec 2024. "Intersecting Families of Graphs", Discrete Mathematics Seminar, Centre for Mathematical Modeling, Santiago Chile.
    • Sep 2024. "Combinatorial Music Theory", DIMAG Lunch Seminar, Institute for Basic Science, Daejeon, Korea.
    • Sep 2024. "Edge-Colored Extremal Problems", Discrete Mathematics Seminar, Institute for Basic Science, Daejeon, Korea.
    • Aug 2024. "The Music and Mathematics of Maximal Evenness in Graphs (presented by co-author Brent Cody), Bridges Conference 2024, Richmond VA.
    • Aug 2024. "Coloring with Forbidden Subgraphs", MAA Mathfest, Indianapolis IN.
    • Oct 2023. "Thresholds for Zero-Sums". AMS Special Session on Extremal and Probabilistic Combinatorics, Fall Sectional Meeting, Mobile, AL.
    • Sep 2023. "Combinatorial Music Theory". VCU Discrete Math Seminar, Richmond VA, USA.
    • Mar 2023. "Even vs. Odd Independent Sets". AMS Special Session on Recent Trends in Structural and Extremal Graph Theory, Atlanta, GA.
    • Mar 2023. "Extremal Problems with Forbidden Color Classes". 54th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL.
    • Oct 2022. "Rainbow Saturated Graphs". Discrete math Seminra, Auburn University, Auburn, AL.
    • Sep 2022. "Threshold Pebbling". VCU Discrete Math Seminar, Richmond VA, USA.
    • Jun 2022. "Partial Rainbows". SIAM Conference on Discrete mathematics. June 2022.
    • Apr 2022. "Bridging the Gap Between Monochrome and Rainbow." AMS Special Session on Topics in Extremal Combinatorics, Joint Mathematics Meetings.
    • Nov 2021. "Rainbow Saturation". Combinatorics Seminar, University of Manitoba, Winnipeg, CAN.
    • Sept 2021. "One of my Favorites: The Sandglass Conjecture via Entropy", VCU Discrete Math Seminar, Richmond, VA.
    • June 2021. "Rainbow Saturation". SIAM Conference on Discrete Mathematics, mini-symposium on Ramsey, Anti-Ramsey, and Extremal Problems.
    • Mar 2021. "Bootstrap Percolation and Automated Conjecturing", 52nd Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL.
    • Feb 2021. "A Gentle Introduction to Extremal Graph Theory", VCU Discrete Math Seminar, Richmond, VA.
    • Mar 2020. "Rainbow Saturation", 51st Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL.
    • Jan 2020. "What is Additive Combinatorics?", VCU Discrete Math Seminar, Richmond, VA.
    • Nov 2019. "Musical Mathematics", VCU Discrete Math Seminar, Richmond VA, USA.
    • Oct 2019. "Variations on a Theme of Turan", Mississippi Discrete Math Workshop, Oxford MS, USA.
    • Oct 2019. "Small Percolating Sets", Mathematics Colloquium, University of Montana, Missoula MT, USA.
    • Aug 2019. "Small Percolating Sets", VCU Discrete Math Seminar, Richmond VA, USA.
    • Mar 2019. "Even Cycles in Dense Graphs", 50th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, USA.
    • Feb 2019. "The Taming of the Hypergraph", VCU Discrete Math Seminar, Richmond VA, USA.
    • Nov 2018. "Extremal Graph Theory", Colloquium, University of Richmond, Richmond VA, USA.
    • Nov 2018. "Automated Conjecturing and Hamiltonian Graphs", MAA Sectional Meeting, University of Mary Washington, Fredericksburg VA, USA.
    • Oct 2018. "Bootstrap Percolation on Infinite Graphs", Mathematics Colloquium, Virginia State University, Petersburg VA, USA.
    • Oct 2018. "Extremal Graph Theory", Mathematics Colloquium, Randolph-Macon College, Ashland VA, USA.
    • Oct 2018. "Bootstrap Percolation on Planar Tilings", Mathematics Colloquium, James Madison University, Harrisonburg VA, USA.
    • Sept 2018. "Bootstrap Percolation on Planar Tilings", Mathematics Colloquium, College of William and Mary, Williamsburg VA, USA.
    • Sept 2018. "Hypergraph Containers Or: How I learned to Stop Worrying and Love Independent Sets", VCU Discrete Math Seminar, Richmond VA, USA.
    • July 2018. "The Even Cycle Spectrum of Dense Graphs", 10th International Colloquium on Graph Theory and Combinatorics, University Lyon 1, La Doua, Lyon, France.
    • June 2018. "Thresholds for Random Pebbling", SIAM Discrete Math, University of Colorado at Denver, Denver, CO, USA.
    • Apr 2018. "Bootstrap Percolation on Polygonal Tilings", MAA Sectional Meeting, Virginia Military Institute, Lexington, VA, USA.
    • Mar 2018, "Automated Conjecturing and Collaborative Mathematics", Forty-Ninth Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, USA.
    • Jan 2018, "2-Connected Graphs Have Many Cycle Lengths", Discrete Math Seminar, Virginia Commonwealth University, Richmond, VA, USA.
    • Sep 2017, "Turan Numbers and their Variants", Discrete Math Seminar, Virginia Commonwealth University, Richmond, VA, USA.
    • Apr 2017, "The Even Cycle Spectrum of Dense Graphs", AMS Special Session on Extremal Problems in Graphs, Hypergraphs and Other Combinatorial Structures, Spring Central Sectional Meeting, Indiana University, Bloomington, IN, USA.
    • Jan 2017, "Variations on a Theme: The Forbidden Subgraph Problem", Mathematics Colloquium, Virginia Commonwealth University, Richmond, VA, USA.
    • Apr 2016, "Minimum Codegree Conditions for Tiling by Tight Cycles", AMS Special Session on Probabilistic and Extremal Combinatorics, AMS Spring North Sectional Meeting, North Dakota State University, Fargo, ND, USA. (Invited)
    • Jan 2016, "Extremal Numbers for Forestable Graphs", AMS-MAA Joint Meetings, Seattle, WA, USA.
    • Nov 2015, "The Forbidden Subgraph Problem and its Variants", Postdoc Lunch Seminar Series, Arizona State University, Tempe, AZ, USA.
    • Oct 2015, "Threshold Pebbling on Grids of Arbitrary Dimension", AMS Special Session on Probabilistic Combinatorics, AMS Fall Southeastern Sectional Meeting, University of Memphis, Memphis, TN, USA. (Invited)
    • Sep 2015, "Pebbling Problems on Graphs", Discrete Math Seminar, Arizona State University, Tempe, AZ.
    • Jun 2015, "Threshold Pebbling for Grids", Connections in Discrete Mathematics, Simon Fraser University, Burnaby, BC, CA.
    • Mar 2015, "Introduction to Hypergraph Containers", Discrete Math Seminar, Arizona State University, Tempe, AZ, USA.
    • Jan 2015, "Supersaturation Theorems, Hypergraph Containers, and Typical Structures", Discrete Math Seminar, Arizona State University, Tempe, AZ, USA.
    • Oct 2014, "Typical Structure of Graphs with No Large Clique", Postdoc Lunch Seminar Series, Arizona State University, Tempe, AZ, USA.
    • Jan 2014, "Random Markov Processes", Mathematics and Computer Science Seminar, Universidade Federal do Ceara, Fortaleza, Brazil.
    • Nov 2013, "Random Markov Processes", Probability Seminar, Arizona State University, Tempe, AZ, USA.
    • Oct 2013, "The Sharp Threshold for Maximum-Size Sum-Free Subsets in Even-Order Abelian Groups", Discrete Math Seminar, Arizona State University, Tempe, AZ, USA.
    • Sep 2013, "Turan Numbers of Linear and Equibipartite Forests", Discrete Math Seminar, Arizona State University, Tempe, AZ, USA.
    • Dec 2012, "Turan numbers of Equibipartite Forests and Forests of Hyperpaths", Theoretical Computer Science and Combinatorics Seminar, Universidade do Sao Paulo, Sao Paulo, Brazil.
    • Dec 2012, "Turan numbers of Linear and Equibipartite Forests", Mathematics and Computer Science Seminar, Universidade Federal do Ceara, Fortaleza, Brazil.
    • Nov 2011, "Turan Numbers for Multiple Paths", Atlanta Lecture Series in Combinatorics and Graph Theory IV, Georgia State University, Atlanta, GA, USA.
    • Oct 2011, "Turan Numbers for Multiple Paths and Some Forests", Discrete Mathematics Seminar, University of Nebraska, Lincoln, NE, USA.
    • Aug 2011, "Turan Numbers for Multiple Paths and Equibipartite Trees", Paul Turan Memorial Conference, Renyi Institute of Mathematics, Budapest, Hungary.
    • Apr 2011, "Turan Numbers for Multiple Paths and Equibipartite Trees", Combinatorics Seminar, University of Memphis, Memphis, TN, USA.
    • May 2008, "Ramsey Theory and Applications", Mathematics Colloquium, Western Washington University, Bellingham, WA, USA.

    Past Outreach Talks:

    • Feb 2022. "Mathematical Music Theory", VCU Math Circle, Virginia Commonwealth University, Richmond, VA. (Audience: High School Students).
    • Nov 2019. "Musical Mathematics", University of Vermont Mathematics Colloquium, Burlington VT.
    • Apr 2019. "Pizza Problems", VCU Math Club, Virginia Commonwealth University, Richmond VA.
    • Dec 2018. "Conway's Rational Tangles", Sonya Kovalevksy Girls in Math Day, Virginia Commonwealth University, Richmond VA. (Audience: Junior High / Middle School Students)
    • May 2018, "Percolation Among Zombies", VCU Math Circle, Virginia Commonwealth University, Richmond, VA. (Audience: High School Students)
    • Apr 2018, "Bootstrap Percolation", Society of Physics Students, Virginia Commonwealth University, Richmond, VA. (Audience: Physics majors and graduate students)
    • Dec 2017, "Conway's Rational Tangles", Sonya Kovalevsky Girls In Math Day, Virginia Commonwealth University, Richmond, VA. (Audience: Junior High / Middle School Students)
    • Apr 2016, "Mad Scientists, Permutations, and Combinatorics", ASU Math Circle, Arizona State University, Tempe, AZ. (Audience: High School Students)
    • Apr 2016, "Conway's Rational Tangles", Mathematics Awareness Day Workshop, Arizona State University, Tempe, AZ, USA. (Audience: High School Students)
    • Sep 2015, "Winning Strategies", ASU Math Circle, Arizona State University, Tempe, AZ. (Audience: High School Students)
    • Sep 2015, "Intro to Extremal Graph Theory", ASU Math Club, Arizona State University, Tempe, AZ. (Audience: Undergraduate Mathematics Majors)
    • Apr 2015, "The Mathematics of Billiards and Reflections", Mathematics Awareness Day Workshop, Arizona State University, Tempe, AZ, USA. (Audience: High School Students. Joint w/ T. Stepien, M. Kawski)
    • Nov 2011, "Voting Theory: Why it isn't fair", Cantor Sect Undergraduate Mathematics Club, University of Memphis, Memphis, TN, USA.