Research

My research interests are in graphy theory and combinatorics. In particular, my main interests are in extremal graph theory and combinatorics. I particularly like questions that are of the following form: "How small can a discrete structure be while still maintaining a particular desired property?" I am forever enamored with the polynomial method. Below is a list of publications and, below that, other activities.
Undergraduate authors are denoted by (u).

    Submissions/To appear


  1. Repeatedly applying the Combinatorial Nullstellensatz for Zero-Sum Grids to Martin Gardner's minimum no-3-in-a-line problem [pdf]
         with Seunghwan Oh(u) and Xianzhi Wang(u)
         to appear in the European Journal of Combinatorics

    Publications


  2. Higher Degree Erdos-Ginzburg-Ziv Constants [pdf]
         with Yair Caro
         Integers: Electronic Journal of Combinatorial Number Theory 22 (2022) A102, 17pp.

  3. Growable Realizations: a Powerful Approach to the Buratti-Horak-Rosa Conjecture [pdf]
         with Matt Ollis, Anita Pasotti, and Marco A. Pellegrini
         Ars Mathematica Contemporanea 22 (2022) 4.04.

  4. Higher Degree Davenport Constants over Finite Commutatitive Rings [pdf]
         with Yair Caro and Benjamin Girard
         Integers: Electronic Journal of Combinatorial Number Theory 21 (2021), A120.

  5. A Survey of Minimum Saturated Graphs (Version 2, updates Version 1 from 2011) [pdf]
          with B.Currie (u), J. Faudree, R. Faudree
                 Electronic Journal of Combinatorics Dynamic Survey 19 (2021), #DS19(Version 2) 98pp.

  6. New Methods to Attack the Buratti-Horak-Rosa Conjecture [pdf]
         with Matt Ollis, Anita Pasotti, and Marco A. Pellegrini
         Discrete Mathematics 344 (2021), no. 9, Paper No. 112486, 20pp.

  7. Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches [pdf]
         with Jacob Hicks and Matt Ollis
         Journal of Combinatorial Designs 27 (2019), no. 6, 369--385.

  8. On Zeros of a Polynomial in a Finite Grid [pdf]
         with Anurag Bishnoi, Pete L. Clark, and Aditya Potukuchi
         Combinatorics Probability and Computing, 27 (2018) 3, 310-333.

  9. Warning's Second Theorem with Restricted Variables[pdf]
           with Pete L. Clark, and Aden Forrow (u)
           Combinatorica 27 (2017) 3, 397-417.

  10. Martin Gardner's minimum no-three-in-a-line problem [pdf]
          with Alec Cooper (u), Oleg Pikhurko and Greg Warrington
           American Mathematical Monthly, 121 (2014), no. 3, 213-221.
           (We note that the published version contains an annoying typographical error in the abstract introduced in the final editing process. The version here eliminates those. And, see the On-line Encyclopedia of Integer Sequences entry A219760.)

  11. On the size and structure of graphs with a constant number of 1-factors [pdf]
          with Andrzej Dudek
           Discrete Mathematics 312 (2012), 1807-1811

  12. Saturation numbers for families of ramsey-minimal graphs [pdf]
          with Guantao Chen, M.Ferrara, R. Gould, and C. Magnant
           Journal of Combinatorics 2 (2011), 435-456.

  13. A Survey of Minimum Saturated Graphs (Version 1, but see Version 2 above!!)[pdf]
          with J. Faudree, R. Faudree
           Electronic Journal of Combinatorics Dynamic Survey 19 (2011), #DS19 36pp.

  14. A Sharp Lower Bound for Potentially H-Graphic Degree Sequences[pdf]
    (Slightly revised from published version thanks to comments from Vera Sos.)
          with M.Ferrara
           SIAM Journal on Discrete Mathematics 23 (2009) 1, 517-526.

  15. Potentially H-Bigraphic Sequences[pdf]
          with M. Ferrara, Mike Jacobson, Mark Siggers
           Discussiones Mathematicae Graph Theory 29 (2009) 583-596.

  16. Using Edge Exchanges to Prove the Erdos-Jacobson-Lehel Conjecture [pdf]       
          with M.Ferrara, Ron Gould
           Bulletin of the Institute of Combinatorics and its Applications 57 (2009), 73-80.

  17. Graphic Sequences with a Realization Containing a Generalized Friendship Graph [pdf]
           with J.H. Yin, Gang Chen
           Discrete Mathematics 308 (2008) 24, 6226-6232.

  18. Graphic Sequences with a Realization Containing a Complete Multipartite Subgraph[pdf]
          with Guantao Chen, M.Ferrara, R. Gould
           Discrete Mathematics 308 (2008) 23, 5712-5721.

  19. Degree Sum Conditions in Graph Pebbling [pdf]
          with Anna Blasiak (u)
           Australasian Journal of Combinatorics 42 (2008), 83-90.
         
  20. A note on minimum K_{2,3}-saturated graphs [pdf]
          with Oleg Pikhurko
           Australasian Journal of Combinatorics 40 (2008), 211-215.
         
  21. Minimum Degree and the Minimum Size of K^t_2-saturated graphs [pdf]
          with R. Gould
           Discrete Mathematics 307 (2007) 9-10, 1108-1114.

  22. Graphic Sequences with a Realization Containing a Friendship Graph [pdf]
          with M. Ferrara, R. Gould
           Ars Combinatoria 85 (2007), 161-171.

  23. Constructive Upper Bounds for Cycle Saturated Graphs of Minimum Size [pdf]
          with Ron Gould, Tomasz Luczak
           Electronic Journal of Combinatorics 13 2006, R29 - 19pp.




An unpublished manuscript

  • Approaching the minimum number of clues Sudoku problem via the polynomial method [pdf]
           with Aden Forrow (u)
          
           (And, read the short story here.)

    Extended Abstracts

  • On the Alon-Furedi bound
           with Anurag Bishnoi, Pete L. Clark, and Aditya Potukuchi
          Electronic Notes in Discrete Mathematics Volume 54, October 2016, 57-62.

  • An Erdos-Stone Type Conjecture for Graphic Sequences [pdf]
          with M.Ferrara
           Electronic Notes in Discrete Mathematics (Proceedings of 6th Czech-Slovak International Symposium, Prague, 2006), Volume 28 (2007), 131-135.
         

    Co-authors

    Anurag Bishnoi, Anna Blasiak, Yair Caro, Alec Cooper, Gang Chen, Guantao Chen, Pete L. Clark, Andrzej Dudek, Jill Faudree, Ralph Faudree, Mike Ferrara, Aden Forrow, Benjamin Girard, Ron Gould, Jacob Hicks, Mike Jacobson, Tomasz Luczak, Colton Magnant, Seunghwan Oh, Matt Ollis, Anita Pasotti, Marco Antonio Pellegrini, Oleg Pikhurko , Aditya Potukuchi , Mark Siggers, Xianzhi Wang, Greg Warrington, J.H. Yin

    Acknowledgements

    I am grateful for support from Fulbright Austria for their invitation to be at U. Graz. I am grateful to Middlebury College for support of my research. I am grateful for past support from the Mathematical Sciences Program at the National Security Agency, and the National Science Foundation, particularly the RUI program, and Vermont EPSCoR.

    Activities

    I was the local organizer for Discrete Mathematics Day of Northeast Conference, held September 20, 2014. In September 2007 and September 2012, I was the local organizer for Discrete Mathematics Day of Northeast Conference, and Discrete Mathematics Day of Northeast Conference.

    In the past, I served (2010-2011) as Secretary of Society of Industrial and Applied Mathematics' Activity Group on Discrete Mathematics.

    Talks

    Here is a list of available recent talks.