Students who I have advised or am advising:
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
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'.
Initially at DeSales University. Now working part time at 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
College of Southern Nevada.
 Ellen
Panofsky, PhD May 2008. `Graph Labeling Problems with
Distance Conditions', Now atTemple 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, 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 St. Vincent College
 Aydin Gerek, Starting Spring 2011
 Caitlin Owens, starting 2015
 Pam Kirkpatrick, starting 2015
 Stash Flowkowski, starting 2016
Postdoctoral visitors to Lehigh that I have worked with:
 Art
Busch 20052006, now at University of Dayton.
 Colton
Magnant, 20082010, now at Georgia Southern University
Here are several unpublished papers:
 My (old, from 1990) PhD dissertation,
Odd forests, reversing numbers, and discrete representations of
interval orders
 Examples of Combinatorial Duality :
an expository paper 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.
 Chain Packings and Odd Subtree Packings :
an old unpublished paper building on work from my PhD thesis. It
seems that part of what I did had been done before and the other
part done several years later by others.
Here are slides from some talks:
Recent talks:
Whats your Order?, Wellesley
College Martha Davenport heard lecture, October 6, 2015
Degrees and trees , Hour talk at 47th
SE Conference on Combinatorics, Graphs Theorey and Computing at
Flordia Atlantic University, March 8, 2016
Choose Multichoose , Hour talk at
47th SE Conference on Combinatorics, Graphs Theorey and Computing
at Flordia Atlantic University, March 11, 2016
Multiset Poker, MAA EPADEL, November
12, 2016
Older talks, Probably pre 2010 but after around 2002 (prior
to that I used overhead slides)
 Perfect maps and factors : My first
computer based talk.
 When does x < a, x > 0 have a
solution? : A talk for undergraduates based on my
expository paper `Examples of combinatorial duality'.
 Score sequences for tournaments
Another talk for undergraduates
 An edge Count Criterion for Degree
Sequences On degree sequences for edge colored graphs
 Intervals and orders: what comes after
interval orders A talk for a special session in memory of
Ken Bogart
 Distinguishing number of cartersian
products of complete graphs
 Structure of bipartite probe interval
graphs
 Star avoiding Ramsey numbers
 Score sequences for tournaments
Another talk for undergraduates
This is an out of date description. The papers also are linked in
the list above.
A research statement from my tenure file in 1998
Here are some of my current research interests and favorite problems. Well  not really; my favorite problems are the hard well known ones. I will mention problems that are less well known and closely related to my current research
Feedback Arc Digraphs: who gets upset in tournament rankings
A feedback arc set in a digraph is a set of arcs whose removal makes the digraph acyclic. A minimum size feedback arc set in a tournament is also the set of arc inconsistent with a `ranking' which minimizes inconsistencies. Every acyclic digraph is a feedback arc set of some tournament. That is, given an acyclic digraph, which we think of as the inconsistencies from a ranking, what is the smallest size of a tournament in which the ranking procedure described above gives exactly those inconsistencies. (We do allow a ranking procedure to produce several equal rankings and we search for the inconsistencies under one of these.) In particular we can look at `the' acyclic tournament as the set of inconsistencies. This provides bounds for other acyclic digraphs. So we ask: what is the smallest size of a tournament which produces a set of n people each pair of which is ranked wrong. I have a specific conjecture about this, along with partial results and related questions in Tournaments as Feedback Arc Sets (Electronic Journal Of Combinatorics Vol 2 (1995) article R20). The partial results are obtained by viewing the problem as an integer linear programming problem. More recent results have been obtained with recent Lehigh PhD Darren Narayan. See his research page for more information.
Powers of Hamiltonian Paths
Label the vertices of a graph so as to maximize the minimum distance between nonadjacent vertices. This is in a sense dual' to the bandwidth problem, which seeks to minimize the maximum distance between adjacent vertices. This problem is also a variant on Hamitonian paths, looking for powers of Hamiltonian paths instead. Some of my papers on this are: Hamiltonian Powers in Threshold and Arborescent Comparability Graphs (with Sam Donnelly, Discrete Mathematics 202 (1999) 3344) and Powers of Hamiltonian Paths in Interval Graphs (Journal of Graph Theory 28 (1998) 3138).
Perfect Maps: how determine your location in n dimensions
List the string 00011101 cyclically. Each triple occurs exactly once. This is known as a perfect map or de Bruijn cycle. Many questions can be asked about higher dimensional versions of this and version with an alphabet of size k (instead of 2). In particular are the `obvious' necessary conditions sufficient? Kenny Patterson in London has answered this completely in 2dimensions for k a prime power. Many other interesting questions arise about related structures called perfect factors and perfect multifactors which arise in the study of these objects. For more information and references to other work see these papers my recent paper (which is partially survey and partially new unifying results and notation Constructing Higher Dimensional Perfect Factors (to appear in Aequationes Mathematicae)
Some recent work on the Prague dimension (product dimension) of graphs with a suprising connection to orthogonal latin squares can be found in Representations of graphs modulo n (with Anthony Evans and Darren Narayan, Discrete Mathematics 223 (2000) 109123)