Garth Isaak's Home Page

Garth Isaak's Home Page

I am an associate professor in the mathematics department at Lehigh University, spouse to psychologist Melissa Hunt and proud parent of Ian Arthur Hunt-Isaak (born 4-8-95) and Noah Cushman Hunt-Isaak (born 11-25-97). This home page mostly has information relevant to my work as a mathematician and educator. For a bit of personal information see the personalized vita below.

For fall 1998 I will be teaching Math/Computer Science 261 - Discrete Structures and Graduate Combinatorics and Math 171 - Problem Solving .

To contact me, send e-mail to gi02@Lehigh.edu, or click here Mail-Me. Click for Lehigh's home page or for Lehigh Mathematics Department home page.

GARTH ISAAK
Department of Mathematics Lehigh University
Bethlehem, PA 18015 USA
Office phone (610)758-3754, Home phone (610)527-4206, e-mail gisaak@lehigh.edu

For Fall 1998 I am teaching Linear Algebra for Business and Economics (Math 61) and a Graduate Graph Theory Course (Math 498).

Here is my family -- I am the larger of the two males pictured, the smaller Ian Arthur Hunt-Isaak (born April 8, 1995) does not yet have his own home page. He is much larger now. My spouse, Melissa Hunt, is the largest of the three females pictured. She has a PhD in clinical psychology from the University of Pennsylvania (where she also won various teching awards), just completed an internship at the Institute of Pennsylvania Hospital and is currently working part time as a therapist at the Coche Center, building hours towards licensing. After thanksgiving she will take a maternity leave after the expected birth of our second child. She does not have a home page, but has an email address (mhunt@cattell.psych.upenn.edu). The dog, Vashti and the cat, Pacem also do not have electronic addresses.

Some combinatorial questions

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, so we ask what is a smallest such tournament. That is, what is the smallest tournament in which the ranking procedure described above 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 (postscript) ( Tournaments as Feedback Arc Sets (dvi) ). The partial result are obtained by viewing the problem as an integer linear programming problem.

Powers of Hamiltonian Paths

Label the vertices of a graph so as to maximize the minimum distance between non-adjacent 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: Powers of Hamiltonian Paths in Interval Graphs; Hamiltonain Powers in Threshold and Arborescent Comparability Graphs

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 2-dimensions for k a prime power. Many other interesting questions arise about related structures called perfect factors and perfect multi-factors which arise in the study of these objects. For more information and references to other work see these papers On Higher Dimensional Perfect Factors ; New Constructions for de Bruijn Tori ; A Meshing Technique for de Bruijn Tori . Constructing Higher Dimensional Perfect Factors is a more recent refernce with a summary of recent work.

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