My Research

I completed my dissertation October 28, 2010 at Western Michigan University.  My adviser is Allen Schwenk.
The k-Cores of a Graph (200 page PDF)

Publications:

  1. Structural Results on Maximal k-Degenerate Graphs, Discussiones Mathematicae Graph Theory, 32 4 (2012) 659-676.
  2. Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
  3. Cores and Shells of Graphs, Mathematica Bohemica. 138 1 (2013) 43-59.
  4. Nordhaus-Gaddum Results for Genus, with Arthur White, Discrete Math, 313 6 (2013) 824-9.
  5. Two Short Proofs on Total Domination, Discuss Math Graph Theory, 33 2 (2013) 457-459.
  6. 2-Tone Coloring of Joins and Products of Graphs, Congr. Num. 217 (2013) 171-190.
  7. Degree Sequences of Monocore Graphs, Discuss Math Graph Theory, 34 3 (2014) 585-592.
  8. Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208
  9. MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
  10. t-Tone Colorings of Graphs, with Benjamin Phillips, Utilitas Math, 106 (2018) 85-102.
  11. Collapsible Graphs, Congr. Num. 231 (2018) 165-172.
  12. Minimum Edge Cuts in Diameter 2 Graphs, with Allen Schwenk, Discuss Math Graph Theory, 39 2 (2019) 605-608.
    Original Article Correction Corrected Article
  13. Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
  14. A Short Proof of Brooks’ Theorem for Vertex Arboricity, AKCE J. of Graphs Combin. (2020) 419-421.
  15. Wiener Indices of Maximal k-Degenerate Graphs, with Zhongyuan Che, Graphs and Combinatorics, 37 2 (2021) 581-589.
  16. Maximal k-degenerate Graphs with Diameter 2, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
  17. k-Paths of k-Trees, Springer PROMS 388 (2020) 287-291.
  18. Properties of Sierpinski Triangle Graphs, 2021. Accepted by Springer PROMS
  19. How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6 (9 pages).
  20. Nordhaus-Gaddum Theorems for Multifactor Decompositions, 2022. Accepted by Springer PROMS
  21. Plane Triangulations Without Spanning 2-Trees, Australas. J. Combin. 85 1 (2023) 82-91.
  22. Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392 (9 pages).
  23. Irregularities of Maximal k-degenerate Graphs, with Zhongyuan Che, Discrete Applied Math. 331 (2023) 70-87.
  24. 2-Tone Coloring of Chordal and Outerplanar Graphs, Australas. J. Combin. 87 1 (2023) 182-197.
  25. A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5 (76 pages).
  26. 2-Tone Coloring of Cactus Graphs, Discrete Applied Math, 347 (2024) 175-186.
  27. Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
  28. Plane Triangulations Without Large 2-Trees, with Gunnar Brinkmann, 2022+.  Submitted.
  29. 2-Tone Coloring of Planar Graphs, 2023+. Submitted.
  30. Comparing Degeneracy and Odd Cycle Bounds for Chromatic Number, 2023+. Submitted.

Expository Math Talks:
The Mathematics of Redistricting
Amazing Arclength
Fun with Pythagorean Triples
MegaMenger Graphs
Graph Classes with Size kn-k(k+1)/2
A New Spin on Cyclic Decompositions
Bounds and Algorithms for Graph Coloring

Research Presentations:
Introduction to the k-Cores of a Graph
k-Degenerate Graphs
Nordhaus-Gaddum Theorems for k-Decompositions
Nordhaus-Gaddum Theorems for k-Decompositions (includes coloring)
2-Tone Coloring and Petersen Covers
New Results on 2-Tone Coloring
2-Tone Coloring of Joins and Products of Graphs
How to Count k-Paths
Maximal Planar Graphs without Large Spanning 2-Trees