Garth Isaak's Mathematics Information

Students who I have advised or am advising:

- Darren Narayan , PhD May 2000. `The Reversing Number of a Digraph'. Now at Rochester Institute of Technology
- Trisha Moller, PhD August 2004. `t-split interval orders'. tPart time a Moravian College and part time with family.
- Brian Heinold, PhD May 2006. `Sum List Coloring and Choosability'. Now at Mt. St. Mary's University Maryland.
- Jen Scancella Gorman, PhD August 2007, `Nested Traveling Salesperson Problems', Now at Lake Superior State University.
- Ellen Panofsky, PhD May 2008. `Graph Labeling Problems with Distance Conditions', Now instructor at Temple University.
- Jonelle Hook, PhD May 2010. `The Clasisfication of Critical graphs and Star-Critical Ramsey Numbers', Now at Mt. St. Mary's University Maryland.
- Breeanne Baker Swart , PhD, May 2013, `The k-Fixed-Endpoint Path Partition Problem', Now at The Citidel.
- Kathleen Ryan, PhD, August 2013, `Degree Sequences of Edge Colored graphs in Specified Families and Related Problems', Now at DeSales University
- Matt Prudente PhD, May 2015, `Two-Player graph Pebbling', Now at Alvernia University
- Aydin Gerek, PhD, January 2017, `Hendry's Conjecture on Chordal graph Subclasses, ' now at Huawei
- Stash Florkowski, PhD, August 2017, `Extensions of Hamiltonian Path and Cycle problems in Special families of Graphs', Now at Center for Army Analysis
- Pam Kirkpatrick Gordon, PhD, August 2018, ` Structural Considerations for Interval orders with Length Constraints', Now at Greenhills School Ann Arbor
- Caitlin Owens, PhD, August 2018, ` Forbidden Subgraphs for Hamiltonian problems on 2-trees', Now at USDA
- Peter Conley, PhD expected July 2023, starting at Penn State Berks
- Jacob Hauser, starting 2021

- Sam Donelley, Undergraduate honors thesis, 1997. `Powers of Hamiltonian paths and Powers of Hamiltonian Cycles in Threshold Graphs'. Now am member of the Database Research Group, University of Pennsylvania.
- Adan Yusko, Undergraduate honors thesis 2010, `Representing Split Graphs as probe Interval graphs', Master degree at Western Michigan, now in industry

Postdoctoral visitors to Lehigh that I have worked with:

- Art Busch 2005-2006, now at University of Dayton.
- Colton Magnant, 2008-2010, now at UPS

I have also served on the PhD thesis committees of a number of students from various departments: computer science, electrical engineering, industrial engineering and mathematics.

Here is a list of my published research papers. I have links to most of the papers (many are prepublication versions nearly in final
form).  For the others if you want a copy please send me an email for a hard copy.

1.G. Isaak and B. Tesman, The Weighted Reversing Number of a Digraph, Congressus Numerantium, 83 (1991) 115-124
2. H. Abeledo and G. Isaak, A Characterization of Graphs that Ensure the Existence of Stable Matchings, Math. Social Sciences, 22 (1991) 93-96
3. G. Isaak, S. Kim, T.A. McKee, F.R. McMorris and F.S. Roberts, 2-Competition Graphs, SIAM J. Discrete Math., 5 (1992) 524-538.
4. G. Isaak, Bounded Discrete Representations of Interval Orders, Discrete Appl. Math., 44 (1993) 157-183.
5. G. Hurlbert and G. Isaak, On the de Bruijn Torus Problem , J. Combinatorial Theory A, 64 (1993) 50-62.
6. G. Hurlbert and G. Isaak, A Meshing Technique for deBruijn Tori , Contemporary Math.  178 (1994) 153-160.
7. G. Isaak, Scheduling Rooted Forests with Communication Delays, Order, 11 (1994) 309-316.
8. J.P. Barthelemy, O. Hudry, G. Isaak, F.S. Roberts and B. Tesman, The Reversing Number of a Digraph , Discrete Appl. Math., 60 (1995) 39-76.
9. K.P. Bogart, P.C. Fishburn, G. Isaak and L. Langley, Proper and Unit Tolerance Graphs , Discrete Appl. Math., 60 (1995) 99-117.
10. G. Hurlbert and G. Isaak, New Constructions for de Bruijn Tori, Designs, Codes and Cryptography, 6 (1995) 47-56.
11. G. Isaak, Tournaments as Feedback Arc Sets , Electronic Journal of Combinatorics, 2 (1995) #R20, 19pp.
12. J. Driscoll, D. Healy and G. Isaak, Scheduling Dyadic Intervals , Discrete Appl. Math., 63 (1995) 101-116.
13. G. Hurlbert and G. Isaak, Equivalence Class Universal Cycles for Permutations , Discrete Math., 149 (1996) 123-129.
14. G. Hurlbert and G. Isaak, On Higher Dimensional Perfect Factors , Ars Combinatoria, 45 (1997) 129-139.
15. K.P. Bogart and G. Isaak, Proper and Unit Bitolerance Orders , Discrete Math., 181 (1998) 37-51.
16. G. Isaak, Powers of Hamilton Paths in Interval Graphs , J. Graph Theory 27 (1998) 31-38.
17. S. Donelley and G. Isaak, Hamiltonian Powers in Threshold and Arborescent Comparability Graphs, Discrete Math., 202 (1999) 33-44.
18. A.B. Evans, G. Isaak and D. Narayan, Representations of Graphs Modulo n, Discrete Math., 223 (2000) 109-123.
19. K.P. Bogart, G. Isaak, J. Liason and A. Trenk, Comparability Invariance Results for Tolerance Orders, Order, 18 (2001) 281-294.
20. G. Isaak, Constructions for Higher Dimensional Perfect Factors , Aequationes Math., 64 (2002) 70-88.
21. G. Isaak, Sum List Coloring 2 by n Arrays, Electron. J. Combinatorics 9 (2002) Note 8, 7 pp.
22. G. Isaak, K. Nyman and A. Trenk, A Heirarchy of Classes of Bounded Bitolerance Orders, Ars Combinatoria, 69 (2003) 33-53.
23. D. Narayan and G.Isaak, A Complete Classifcation of Tournaments having a Disjoint Union of Directed Paths as a Minimum Feedback Arc Set , J. Graph Theory, 45 (2004) 28 - 47.
24.G. Isaak Sum List Coloring Block Graphs, Graphs and Combinatorics 20 (2004) 499-506.
25. D. Narayan and G. Isaak, A Classification of Tournaments having an Acyclic Tournament as a Minimum Feedback Arc Set , Inf. Proc. Let. 92 (2004) 107-111.
26. G. Isaak, Hamiltonicity of Digraphs for Universal Cycles of Permutations, European Journal of Combinatorics 27 (2006) 801-805.
27. A.H. Busch and G. Isaak, Recognizing Bipartite Unbounded Tolerance Graphs in Linear Time, Lecture Notes in Computer Science 4769 (2007) 12-20 .
28. M.J. Fisher and G. Isaak, Distinguishing Colorings of Cartesian Products of Complete Graphs, Discrete Mathematics 308 (2008) 2240-2246.
29. G. Isaak, Interval Order representation via Shortest Paths, in The Mathematrics of Preference, Choice and Order - Essays in honor of Peter Fishburn,
S. Brams, W.V. Gehrlein, F.S. Roberts eds, Springer (2009) 303-311.
30. G. Isaak, R. Jamison, D. Narayan, Greedy Rankings and Arank Numbers, Information Processing Letters 109 (2009) 825-827.
31. D.E. Brown, A.H. Busch, G. Isaak, Linear Time Recognition Algorithms for Bipartite Tolerance graphs and Bipartite probe Interval graphs, Discrete Mathematics and Theoretical Computer Science, 10 (2010) 63-82.
32. G. Isaak and D.B. West, The Edge-count Criterion for Graphic Lists, Electronic Journal of Combinatorics, 17 (2010) N36, 5 pp.
33. J. Hook and G. Isaak, Star-Critical Ramsey Numbers ,  Discrete Applied Mathematics 159 (2011) 328-334.
34. M.J. Fisher and G. Isaak, Distinguishing numbers of Cartesian products of multiple complete graphs, Ars Mathematica Contemporanea 5 (2012) 159-173.
35. G. Isaak, P. Johnson, C. Petrie, Integer and Fractional Security in graphs, Discrete Applied Mathematics 160 (2012) 2060-2062.
36. S. Boyadzhiyska, G. Isaak, A. Trenk, A Simple Proof Characterizing Interval Orders with Interval Lengths Between 1 and k, Involve 11 (2018) 893-900.
37. G. Isaak, An Alternate Approach to Changing Sine, Mathematics Magazine 91 (2018) 337-340.
38. S. Boyadzhiyska, G. Isaak, A. Trenk, Interval Orders with two interval Lengths, Discrete Appl. Math 267 (2019) 52-63.
39. G. Isaak and M. Prudente, Two-Player Pebbling on Diamter 2 Graphs, Internat. J. Game Theorey 50 (2021) 581-596.
40. G. Isaak, M. Prudente, A. Potylycki, W. Fagley, J. Marcinik, On Two-Player Pebbling, Communications on Number Theory and Combinatorial Theory, 3 (2022).
41. G. Isaak, M. Prudente, J. M. Marcinik III, A Pebbling Game on Powers of Paths Communications on Number Theory and Combinatorial Theory, 4 (2023).

Some `recent' unpublished expository manuscripts (from 2016-2018 but working on more recent updates in 2023)
- G. Isaak, Degrees and 2-Multitrees: Characterizes degree sequences of 2-multitrees
- G. Isaak, Degree Sequences for Multitrees: Describes Degree sequences of multitrees. Includes my favorite example of a nontrivial failed basis for induction.
- G. Isaak, L. Langley Catalan Multijections : Examines various proofs for Catalan Numbers with a notion of what it means for them to be different.
- G. Isaak Examples of Combinatorial Duality : an expository paper from around 2005 giving examples of the theorem of the alternative for linear
inequalities is used to prove Hoffman's circulation theorem and then from this Fishburn's Interval order representation
theorem and Landau's theorem for score sequences of tournaments. Portions now appear in my published paper Interval Order representation via Shortest Paths.

Some old unpublished research manuscripts

- J. Carroll and G. Isaak, Degree Matrices Realized by Edge Colored Forests , manuscript, March 2010.
- Chain Packings and Odd Subtree Packings : around 1991, an old unpublished paper building on work from my PhD thesis

My old Dissertation and undergraduate theses plus some unpublished research papers:
- My 1990 PhD dissertation from RUTCOR (Rutgets center for Operations Research, Odd forests, reversing numbers, and discrete representations of interval orders
- My undergraduate 1984 senior thesis in Mathematics at Bethel College, Kansas on Godel's Theorem A `story' attmpting to illustrate Godel's theorem
- My underrgaduate 1984 senior thesis in Physics and Peace Studies at Bethel College, Kansas What it means to Count: Issues of Verification in Strategic Arms Treaties
An examination of Nuclear Arms races in the early 1980's using Richardson's model and a new interpretation of how to measure arms strenth in that model.

Some `recnt talks' (last 10 years), many about as yet unpublished work

- Enumerating List Colorings March 2023
- Catalan Multijections October 2019
- Forbidden SEO-Induced Subgraphs
- Voting Profiles, Constant Row Tomography, Edge Coloring Regular Bipartite Multigraphs, 3-Dimensional Contingency Table Reconstruction, Birkhoff-Non neumann Theorem, ... November 2018
- A Two Player graph Pebbling Game September 2018
- Hamiltonian Path Variants in Structured Graph Families June 2018
- Interval orders with Length Bounds March 2017
- Multiset Poker March 2015
- Min-Max theorems for the k-Path Partition Problem (in highly structured graphs) May 2014
- Degree Lists for Multiforests and Near Multiforests January 2014
- Degree Lists for partial 2-Trees March 2013
- Cycle Extendability in Ptolemaic Graphs May 2012

Here are slides from some longer (and older) talks, many expository for undergraduate audiences:

- Perfect maps and factors : My first computer based talk in 2004 at Midwest Conference on Combinatorics and Combinatorial Computing, Rochester Institute of Technology October 2004 (prior to that used overhead slides).
- Intervals and orders: what comes after interval orders A talk for a SIAM Discrete Math special session in memory of Ken Bogart, June 2006
- When does x < a, x > 0 have a solution? : A talk for undergraduates based on my expository paper `Examples of combinatorial duality, given several times around 2010
- Score sequences for tournaments Another related talk for undergraduates given several times around 2010
- 10 Binomial (and other) facts A talk for general audience about the binomial triangle
- Mathematics of Round Robin Tournaments A general audience talk at my undergraduate college, Bethel, Kansas, October 2011. Includes some old pictures of me and other old historical notes
- Boundaries for Graph Ramsey Numbers Swarthmore College fall 2013.
- Whats your Order?, Wellesley College Martha Davenport Heard lecture, October, 2015
- Degrees and trees , Hour talk at 47th SE Conference on Combinatorics, Graph Theory and Computing at Flordia Atlantic University, March, 2016
- Choose Multichoose , Hour talk at 47th SE Conference on Combinatorics, Graph Theory and Computing at Flordia Atlantic University, March, 2016

Older Talks

- Degrees in Edge Colored Graphs
- Star avoiding Ramsey numbers
- Structure of bipartite probe interval graphs
- Distinguishing number of cartersian products of complete graphs
- An edge Count Criterion for Degree Sequences

Garth Isaak home page