David Galvin's Research


My research is in discrete probability, combinatorics and graph theory; in particular, extremal enumerative problems, set partitions, and the applications of combinatorial ideas to the study of phase transitions in statistical physics models and long-range correlations in discrete random structures. You can find a 2012 summary of my research here. Below are links to my papers (not necessarily the final versions); most of these can also be found at arXiv.org.

For links to journal versions of these papers, please go here; note that a Notre Dame netID is needed to access this page.

Publications

    In preparation

  1. Ordered partitions and the thinned exponential function (with J. Engbers and C. Smyth), not yet available.
  2. Under revision

  3. Total non-negativity of generalized Stirling matrices (with A. Pacurar)
  4. Submitted

  5. Independent set and matching permutations (with T. Ball, K. Hyry and K. Weingartner)
  6. Cutting lemma and Zarankiewicz's problem in distal structures (with A. Chernikov and S. Starchenko)
  7. 2019

  8. Phase Coexistence for the Hard-Core Model on Z^2 (with A. Blanca, Y. Chen, D. Randall and P. Tetali), Combinatorics, Probability and Computing, 28 (2019), 1-22. The data sets used in this paper are available here
  9. The game of plates and olives (with T. Carroll), Electronic Journal of Combinatorics 26 (2019), P1.18
  10. Restricted Stirling and Lah number matrices and their inverses (with J. Engbers and C. Smyth), Journal of Combinatorial Theory Series A 161 (2019), 271-298
  11. Independent sets in the discrete hypercube (an expository article)
  12. 2018

  13. The independent set sequence of some families of trees (with J. Hilyard), Australasian Journal of Combinatorics 70 (2018), 236-252
  14. 2017

  15. Extremal H-colorings of trees and 2-connected graphs (with J. Engbers), Journal of Combinatorial Theory Series B 122 (2017), 800-814
  16. 2016

  17. On the independence ratio of distance graphs (with J. Carraher, S. Hartke, J. Radcliffe and D. Stolee), Discrete Mathematics 339 (2016), 3058-3072
  18. Asymptotic normality of some graph sequences, Graphs and Combinatorics 32 (2016), 639-647
  19. 2015

  20. Phase coexistence and torpid mixing in the 3-coloring model on Z^d (with J. Kahn, D. Randall and G. Sorkin), SIAM Journal of Discrete Mathematics 29 (2015), 1223-1244
  21. Counting colorings of a regular graph, Graphs and Combinatorics 31 (2015), 629--638
  22. Combinatorially interpreting generalized Stirling numbers (with J. Engbers and J. Hilyard), European Journal of Combinatorics 43 (2015), 32--54
  23. 2014

  24. Three tutorial lectures on entropy and counting (notes prepared to accompany a series of tutorial lectures given at the 1st Lake Michigan Workshop on Combinatorics and Graph Theory, Western Michigan University, March 15--16 2014), arXived
  25. Counting independent sets of a fixed size in graphs with given minimum degree (with J. Engbers), Journal of Graph Theory 76 (2014), 149--168
  26. 2013

  27. Phase Coexistence and Slow Mixing for the Hard-Core Model on Z^2 (with A. Blanca, D. Randall and P. Tetali), Lecture Notes in Computer Science 8096 (Proceedings of APPROX/RANDOM 2013) (2013), 379--394. The data sets used in this paper are available here
  28. Stirling numbers of forest and cycles (with Do Trong Thanh), Electronic Journal of Combinatorics 20 (2013), #P73
  29. Maximizing H-colorings of regular graphs, Journal of Graph Theory 73 (2013), 66--84
  30. 2012

  31. H-coloring tori (with J. Engbers), Journal of Combinatorial Theory Series B 102 (2012), 1110-1133
  32. The independent set sequence of regular bipartite graphs, Discrete Mathematics 312 (2012), 2881--2892
  33. H-colouring bipartite graphs (with John Engbers), Journal of Combinatorial Theory Series B 102 (2012), 726-742
  34. Reverse Mathematics and infinite traceable graphs (with P. Cholak and R. Solomon), Mathematical Logic Quarterly 58 (2012), 18--28
  35. 2011

  36. Two problems on independent sets in graphs, Discrete Mathematics 311 (2011), 2105--2112
  37. The multi-state hard core model on a regular tree (with Fabio Martinelli, Kavita Ramanan and Prasad Tetali), SIAM Journal of Discrete Mathematics 25 (2011), 894--916
  38. The number of independent sets in a graph with small maximum degree (with Yufei Zhao), Graphs and Combinatorics 27 (2011), 177--186. The java programs used in this paper (source files, compiled programs and documentation) are available here in a compressed (zipped) folder.
  39. A threshold phenomenon for random independent sets in the discrete hypercube, Combinatorics, Probability and Computing 20 (2011), 27--51
  40. 2002--2010

  41. An upper bound for the number of independent sets in regular graphs, Discrete Mathematics 309 (2009), 6635--6640
  42. Matchings and Independent Sets of a Fixed Size in Regular Graphs (with Teena Carroll and Prasad Tetali), Journal of Combinatorial Theory Series A 116 (2009), 1219--1227)
  43. Sampling independent sets in the discrete torus, Random Structures and Algorithms 33 (2008), 356--376
  44. Sampling 3-colourings of regular bipartite graphs, Electronic Journal of Probability 12 (2007), 481--497
  45. Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus (with Dana Randall), Proceedings of the Eighteenth Annual ACM--SIAM Symposium on Discrete Algorithms (2007), 376--384
  46. Bounding the partition function of spin systems, Electronic Journal of Combinatorics 13 (2006), #R72
  47. Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs (with Prasad Tetali), Random Structures and Algorithms 28 (2006) 427--443
  48. Global connectivity from local geometric constraints for sensor network with various wireless footprints (with Raissa D'Souza, Cris Moore and Dana Randall), Proceedings of the Fifth International Conference on Information Processing in Sensor Networks (2006), 19--26
  49. On weighted graph homomorphisms (with Prasad Tetali), DIMACS series in discrete mathematics and theoretical computer science 63 (Graphs, Morphisms and Statistical Physics) (2004), 97--104
  50. On phase transition in the hard-core model on Z^d (with Jeff Kahn), Combinatorics, Probability and Computing 13 (2004), 137--164
  51. Slow mixing of the Glauber Dynamics for the hard-core model on the Hamming cube (with Prasad Tetali), Proceedings of the Fifteenth Annual ACM--SIAM Symposium on Discrete Algorithms (2004), 459--460
  52. Entropy and graph homomorphisms, Oberwolfach reports 1 (2004), 30--32
  53. On homomorphisms from the Hamming cube to Z, Israel Journal of Mathematics 138 (2003), 189--213
  54. Two problems involving the notion of phase transition (PhD thesis written under the direction of Jeff Kahn, Rutgers University, October 2002), ProQuest Dissertations and Theses (2002)

Miscellaneous other papers

  1. Ramanujan Graphs (Essay for Part III of Mathematical Tripos, University of Cambridge, June 1996)
  2. [4]^2 Tic-Tac-Toe is a draw
  3. Erdos's proof of Bertrand's postulate (an exposition of Erdos's elementary proof that there is always a prime between n and 2n)
  4. Ultrafilters, with applications to analysis, social choice and combinatorics (an introduction to the basics of ultrafilters, with some applications)
  5. A topological approach to evasiveness (an introduction to the use of topological ideas in the study of evasive properties, based on a paper by Kahn, Saks and Sturtevant)
  6. Bregman's theorem and extensions (an exposition of Radhakrishnan's entropy proof of Bregman's theorem, and some results and conjectures around the theorem)

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