My Research

A look at my mathematical interests

In case you're curious, here's some of the fruits of my mental labour of the past few years.

A rough statement of interests

My main area of research is graph theory: the study of discrete structures made up of vertices (representing some set of objects) and edges (representing a symmetric binary relationship between those objects). More specifically, I've worked in fractional graph theory, network connectivity, graph colouring, and graph decompositions; this last might be more properly considered to be design theory, the mathematics of discrete structures with certain balance properties.

Other things which fascinate me: I'm very interested in the design of tournaments and league schedules, and the underlying design theory methods that are used. I'm also interested in applications of discrete mathematics in the social sciences: this includes voting theory in its several strains, problems of fair division, game theory, and the theory of social networks.

My papers

What follows is a list of my articles, both those that have been published and... otherwise. If you're interested in [p]reprints, feel free to contact me and ask.

Published papers

Another Network Vulnerability Parameter

(With P. D. Johnson Jr.; appeared in Congressus Numerantium 153.) My first published paper, and also the first one that I had a hand in writing. It treats the problem that I based my dissertation on... something about networks with unstable vertices. This paper (and my presentation of it, at a conference in March of 2001) has also given my some minor notoriety in certain circles, because of the story I use as an example of an application...

A note on the Grundy number

(Appeared in Bulletin of the ICA 38.) My first solo effort and a whopping four pages. Essentially pointing out and filling a hole in a twenty-year-old paper by a bunch of people who have been doing this stuff longer than I have.

Concerning fractional automorphisms

(With P. D. Johnson Jr. and Robert Rubalcaba; appeared in Congressus Numerantium 159.) Rob was my roommate during my last year-and-a-bit of grad school; we started working on this stuff when he was avoiding working on what was supposed to be his Masters' project; this work formed the foundation for his Doctoral dissertation.

The competition chromatic numbers of a graph

(Appeared in Congressus Numerantium 164.) This is the sort of thing I go to conferences for. Mere minutes before Rob and I hopped in his truck to brave the 11+ hour trip from south Florida to central Alabama, I heard a talk about competition graph parameters. I immediately saw a way to twist the idea around to talk about graph colourings, babbled something (no doubt incoherent) at the speaker after his talk was over, and was gone.

The actions of fractional automorphisms

(With P. D. Johnson Jr. and Robert Rubalcaba; appeared in JCMCC 50.) The sequel to Concerning fractional automorphisms, above. We extended the results, answered our major open question from the first paper, and brought the terminology more into line with current usage.

Domination null and packing null vertices

(With P. D. Johnson Jr. and Robert Rubalcaba; appeared in Congressus Numerantium 168.) Another product of the Johnson Family School of Fractional Graph Theory. Less of a hard-core research paper than the beginning of a dialogue.

The hub number of a graph

(Appeared in the International Journal of Mathematics and Computer Science 1.) Like about two-thirds of the research I do, it's about a concept that's so totally out of left field that it hadn't occured to anyone to think of it before. I've presented this one three times so far, and the third time was indeed the charm since it landed me my current position. I was invited to submit a paper for the inaugural issue of this particular journal, and this seemed like a good one.

Even spanning trees in bipartite graphs

(With Dean G. Hoffman; appeared in the Australasian Journal of Combinatorics 35.) Technically my first paper, although something like my eighth to be published. A speaker mentioned an open problem at a conference in South Carolina; I came up with an answer that looked good but couldn't quite prove it. I shared my conjecture with Dean Hoffman over some bad American beers; three days later he came up to me with a lovely matroid-theoretic proof.

Accepted Papers

Fractional inverse and inverse fractional domination

(With P. D. Johnson Jr.; accepted to Ars Combinatoria.) There's a moderately notorious problem about the inverse domination number that I first heard about at the same conference that spawned Even spanning trees; this paper solves the fractional analogue of that problem. Well, both such analogues, really.

A characterization of lattice-ordered graphs

(With C. D. Leach; accepted to Integers: the on-line journal of combinatorial number theory.) One goes to conferences in order to confer; seeing other people's presentations is important but secondary. David and I started to discuss this problem as I was driving him to the Fort Lauderdale airport during/after the Southerastern Conference one year; we pretty much worked out the solution over the course of the next couple of months. In case you're curious, graph are lattice-ordered when their subgraphs relate to each other in certain aesthetically pleasing ways, and it doesn't happen very often.

The chromatic villainy of a graph

(With S. A. Clark, S. H. Holliday, J. E. Holliday, P. D. Johnson Jr., Robert Rubalcaba, and J. E. Trimm; accepted to Congressus Numerantium.) Wow, that's a lot of co-authors. A bunch of us got together in northwestern Tennessee for a weekend, to visit and fondue and ostensibly get some work done. This turned out to be the work; the villainy refers to how badly a proper colouring of a graph can get screwed up.

Papers being refereed

Minimum fractional dominating and maximum fractional packing functions

(With Robert Rubalcaba; submitted.) I used to call this The Paper that Wouldn't Die, because of the absurdly long time it was taking to assemble. My first year out of Auburn, Rob and I met up in Nashville, roughly midway between our respective homes; the basic insight that drives this paper was devised then. We had most of the major results by the following November; it took another year before we got around to writing it all up.

Routing sets in the integer lattice

(With P. Hamburger and R. Vandell; submitted.) Started as an extension of the hub number ideas, developed into something a little bit different. This paper's been through two or three name changes as it expanded and mutated. My most successful conference talk so far was about this material.

(a,b)-domination: a unified linear framework for some domination parameters

(With A. Abueida and Robert Rubalcaba; submitted.) Coming up with graph parameters that look sort of like domination is something of a cottage industry. It's often useful to find a general framework, a template for the different parameters, which lets you draw both general conclusions about the whole family and specific relationships between members of the family. One approach is via integer linear programming, and that's what we're pursuing here.

p-norm fractional domination in graphs

(With D. G. Hoffman, P. D. Johnson Jr., and Robert Rubalcaba; submitted.) I'm pretty sure this one started out as poolside chatter in Florida. Fractional domination is a game of minimization: you want the smallest possible function with a given property. If you treat the functions as vectors (and there's no reason not to), then there's some ambiguity about exactly how to measure the smallness of a function. Standard domination theory uses one way; here we're considering a whole bunch of others. Notable because I get to cite work done by my department chair and associate chair, who work in a completely different branch of mathematics than I do.

Papers in Limbo

Fuzzification and its discontents: a case study

(With P. D. Johnson Jr. and Robert Rubalcaba; in revisions.) This originated from a bit of idle chatter in Auburn's local brewpub, about why something was defined the way it was. The answer turned out to be: because the other possible definitions lead to silly, silly results. This is their story. We've submitted this one twice and had it rejected twice, perhaps because the actual results are not terrifically interesting mathematically; we're trying to clean it up a bit to make our point more clearly.

The maximum dimension of a shallow graph

(In revisions.) This one's still a bit of a work in progress. I came up with the basic idea while still a graduate student, and worked out the solution to the question I'd posed. I then proceeded to stick it in a drawer for years in hopes of figuring out what it's good for. All of the methods are elementary, which means that I got to stick it on a final exam for a graph theory course with impunity.

Invariants of fractional isomorphism

(With Robert Rubalcaba; in progress.) Starting from our earlier observation that if you treat a fractional automorphism as an operator on vectors, then it maps fractional dominating functions to fractional dominating functions, we came up with the idea for this paper. Fractional isomorphism means that two graphs look the same in a terribly broad kind of way; we're looking at exploring exactly how not-alike two fractional isomorphs can be.

Nonseparable and irreducible Hamilton decompositions

(With S. H. Holliday; in progress.) Sarah shoved a problem at me as we were sitting down to hear a talk at the Southeastern Conference; over the course of the next couple of weeks we got some nice-looking constructions, but somehow we've never quite managed to finish writing all of this up. Soon, perhaps.

Fractional domination in prisms

(In progress.) Apparently the second in a series of papers where I establish the fractional version of an integer conjecture. This has possibly had the shortest genesis of any paper of mine, since I went from hearing about the original question to having a mostly-completed draft in something like three weeks.