Research

I am seeking sabbatical opportunities in combinatorics for the academic year 2015-16. Please send me your suggestions!

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 have recently become enamored with the polynomial method. Below is a list of publications and, below that, other activities.
Undergraduate authors are denoted by (u).

In preparation/Submissions/To appear

  1. Warning's Second Theorem with Restricted Variables[pdf]
           with Pete L. Clark, and Aden Forrow (u)
           submitted

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

    Publications

  3. 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.)

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

  5. 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.

  6. A Survey of Minimum Saturated Graphs [pdf]
          with J. Faudree, R. Faudree
           Electronic Journal of Combinatorics Dynamic Survey 18 (2011), #DS19 36pp.

  7. 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.

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

  9. 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.

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

  11. 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.

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

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

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




Extended Abstract

  • 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

    Anna Blasiak, Alec Cooper, Gang Chen, Guantao Chen, Pete L. Clark, Andrzej Dudek, Jill Faudree, Ralph Faudree, Mike Ferrara, Aden Forrow, Ron Gould, Mike Jacobson, Tomasz Luczak, Colton Magnant, Oleg Pikhurko , Mark Siggers, J.H. Yin, Greg Warrington

    Acknowledgements

    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 am the local organizer for Discrete Mathematics Day of Northeast Conference, to be held September 15, 2012. In September 2007, I was the local organizer for 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.