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).

- Minimum no-k-in-a-line problem under unique representation[pdf]

submitted

- On Zeros of a Polynomial in a Finite Grid[pdf]

with Anurag Bishnoi, Pete L. Clark, and Aditya Potukuchi

submitted

- Warning's Second Theorem with Restricted Variables[pdf]

with Pete L. Clark, and Aden Forrow (u)

to appear in Combinatorica

- 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

- 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.)

- On the size and structure of graphs with a constant number of 1-factors [pdf]

with Andrzej Dudek

Discrete Mathematics**312**(2012), 1807-1811

- 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.

- A Survey of Minimum Saturated Graphs [pdf]

with J. Faudree, R. Faudree

Electronic Journal of Combinatorics Dynamic Survey 18 (2011), #DS19 36pp.

- 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.

- Potentially H-Bigraphic Sequences[pdf]

with M. Ferrara, Mike Jacobson, Mark Siggers

Discussiones Mathematicae Graph Theory**29**(2009) 583-596.

- 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.

- Graphic Sequences with a Realization Containing a Generalized Friendship Graph
[pdf]

with J.H. Yin, Gang Chen

Discrete Mathematics**308**(2008) 24, 6226-6232.

- 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.

- Degree Sum Conditions in Graph Pebbling [pdf]

with Anna Blasiak (u)

Australasian Journal of Combinatorics**42**(2008), 83-90.

- A note on minimum K_{2,3}-saturated graphs [pdf]

with Oleg Pikhurko

Australasian Journal of Combinatorics**40**(2008), 211-215.

- Minimum Degree and the Minimum Size of K^t_2-saturated graphs
[pdf]

with R. Gould

Discrete Mathematics**307**(2007) 9-10, 1108-1114.

- Graphic Sequences with a Realization Containing a Friendship Graph
[pdf]

with M. Ferrara, R. Gould

Ars Combinatoria**85**(2007), 161-171.

- Constructive Upper Bounds for Cycle Saturated Graphs of Minimum
Size [pdf]

with Ron Gould, Tomasz Luczak

Electronic Journal of Combinatorics**13**2006, R29 - 19pp.

with M.Ferrara

Electronic Notes in Discrete Mathematics (Proceedings of 6th Czech-Slovak International Symposium, Prague, 2006), Volume 28 (2007), 131-135.

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