Garth Isaak's Mathematics Information
Students who I have advised or am advising:
Graduate:
 Darren Narayan ,
PhD May 2000. `The Reversing Number of a Digraph'. Now at
Rochester Institute of Technology
 Trisha Moller, PhD August 2004. `tsplit 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 StarCritical Ramsey Numbers', Now at Mt. St. Mary's
University Maryland.
 Breeanne
Baker Swart , PhD, May 2013, `The kFixedEndpoint
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, `TwoPlayer 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 2trees', Now at USDA
 Peter Conley, PhD expected July 2023, starting at Penn State Berks
 Jacob Hauser, starting 2021
Undergraduate:
 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 20052006, now at University of Dayton.
 Colton
Magnant, 20082010, 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) 115124
2. H. Abeledo and G. Isaak, A Characterization
of Graphs that Ensure the Existence of Stable Matchings, Math.
Social Sciences, 22 (1991) 9396
3. G. Isaak, S. Kim, T.A. McKee, F.R. McMorris and F.S. Roberts, 2Competition Graphs, SIAM J. Discrete
Math., 5 (1992) 524538.
4. G. Isaak, Bounded Discrete Representations
of Interval Orders, Discrete Appl. Math., 44 (1993) 157183.
5. G. Hurlbert and G. Isaak, On the de Bruijn
Torus Problem , J. Combinatorial Theory A, 64 (1993) 5062.
6. G. Hurlbert and G. Isaak, A Meshing
Technique for deBruijn Tori , Contemporary Math. 178
(1994) 153160.
7. G. Isaak, Scheduling Rooted Forests with
Communication Delays, Order, 11 (1994) 309316.
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) 3976.
9. K.P. Bogart, P.C. Fishburn, G. Isaak and L. Langley, Proper and Unit Tolerance Graphs , Discrete
Appl. Math., 60 (1995) 99117.
10. G. Hurlbert and G. Isaak, New
Constructions for de Bruijn Tori, Designs, Codes and
Cryptography, 6 (1995) 4756.
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)
101116.
13. G. Hurlbert and G. Isaak, Equivalence
Class Universal Cycles for Permutations , Discrete Math., 149
(1996) 123129.
14. G. Hurlbert and G. Isaak, On Higher
Dimensional Perfect Factors , Ars Combinatoria, 45 (1997)
129139.
15. K.P. Bogart and G. Isaak, Proper and Unit
Bitolerance Orders , Discrete Math., 181 (1998) 3751.
16. G. Isaak, Powers of Hamilton Paths in
Interval Graphs , J. Graph Theory 27 (1998) 3138.
17. S. Donelley and G. Isaak, Hamiltonian Powers
in Threshold and Arborescent Comparability Graphs, Discrete
Math., 202 (1999) 3344.
18. A.B. Evans, G. Isaak and D. Narayan,
Representations of Graphs Modulo n, Discrete Math., 223 (2000)
109123.
19. K.P. Bogart, G. Isaak, J. Liason and A. Trenk, Comparability Invariance Results for Tolerance
Orders, Order, 18 (2001) 281294.
20. G. Isaak, Constructions for Higher
Dimensional Perfect Factors , Aequationes Math., 64 (2002)
7088.
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) 3353.
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) 499506.
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) 107111.
26. G. Isaak, Hamiltonicity of Digraphs for
Universal Cycles of Permutations, European Journal of
Combinatorics 27 (2006) 801805.
27. A.H. Busch and G. Isaak, Recognizing Bipartite
Unbounded Tolerance Graphs in Linear Time, Lecture Notes in
Computer Science 4769 (2007) 1220 .
28. M.J. Fisher and G. Isaak, Distinguishing
Colorings
of Cartesian Products of Complete Graphs, Discrete
Mathematics 308 (2008) 22402246.
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) 303311.
30. G. Isaak, R. Jamison, D. Narayan, Greedy
Rankings and Arank Numbers, Information Processing Letters 109
(2009) 825827.
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) 6382.
32. G. Isaak and D.B. West, The Edgecount Criterion for Graphic
Lists, Electronic Journal
of Combinatorics, 17 (2010) N36, 5 pp.
33. J. Hook and G. Isaak, StarCritical Ramsey
Numbers , Discrete Applied Mathematics 159 (2011)
328334.
34. M.J. Fisher and G. Isaak, Distinguishing
numbers of Cartesian products of multiple complete graphs, Ars
Mathematica Contemporanea 5 (2012) 159173.
35. G. Isaak, P. Johnson, C. Petrie, Integer and
Fractional Security in graphs, Discrete Applied Mathematics
160 (2012) 20602062.
36. S. Boyadzhiyska, G. Isaak, A. Trenk, A Simple Proof
Characterizing Interval Orders with Interval Lengths Between 1 and
k, Involve 11 (2018) 893900.
37. G. Isaak, An Alternate Approach to Changing Sine, Mathematics
Magazine 91 (2018) 337340.
38. S. Boyadzhiyska, G. Isaak, A. Trenk, Interval Orders with two interval Lengths, Discrete
Appl. Math 267 (2019) 5263.
39. G. Isaak and M. Prudente, TwoPlayer Pebbling on Diamter 2 Graphs,
Internat. J. Game Theorey 50 (2021) 581596.
40. G. Isaak, M. Prudente, A. Potylycki, W. Fagley, J. Marcinik, On TwoPlayer 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 20162018 but working on more recent updates in 2023)
 G. Isaak, Degrees and 2Multitrees:
Characterizes degree sequences of 2multitrees
 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 SEOInduced Subgraphs
 Voting Profiles, Constant Row Tomography, Edge Coloring Regular Bipartite Multigraphs,
3Dimensional Contingency Table Reconstruction, BirkhoffNon 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
 MinMax theorems for the kPath Partition Problem (in highly structured graphs) May 2014
 Degree Lists for Multiforests and Near Multiforests January 2014
 Degree Lists for partial 2Trees 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