Recently I had a paper rejected by a journal including the words “discrete mathematics” in its title. The paper was sent back unrefereed, the editor explaining that in their judgment it was group theory not discrete mathematics. In fact the editor was wrong on several counts, which I will come to.
A lot of my work is situated between groups (usually finite) and discrete mathematics. When I post a paper on the arXiv, I have to decide whether the primary classification is Group Theory or Combinatorics. It quite often happens that the arXiv robot disagrees; in these circumstances, it is possible to disregard the robot, but it is probably easier to go along with it. There is no question of the paper being rejected, and a few screens later you can give a secondary classification. But having a paper instantly rejected by a journal is another matter.
So why do I say that the editor was wrong?
First, readers will not be surprised to learn that the paper was about a certain type of graph defined from a group. Unlike perhaps my usual approach to such things, it treated the graph simply as a graph, and the results proved had no implications for group theory. Moreover, there is no group theory at all in the paper: all we needed to know about the graphs had already been provided by group theorists (and graph theorists).
Second, the paper contained a result in pure graph theory, which I have already discussed here, as it happens. I regard this as one of the highlights of the paper, and it certainly has other uses, which would earn the paper a citation.
But the main reason for my concern is my deeply held belief that finite group theory, at least, is a branch of discrete mathematics. I divide mathematics into discrete and continuous (prickles and goo, as Alan Watts put it), and finite group theory is certainly not continuous mathematics: even character theory, coming from representations over the complex numbers, can be done without detriment in finite extensions of the rationals.
I will go on a bit longer, since I think there is more to be said. I have quoted many times the celebrated theorem of Brauer and Fowler that there are only finitely many finite simple groups with a given involution centraliser, which opened the door to the classification of finite simple groups. Although the word “graph” doesn’t appear in the paper, they clearly used graph theoretic methods in the argument, which relied on the distance in the commuting graph.
There used to be a subject called “Combinatorial group theory”. It is now more usually known as “Geometric group theory”, reflecting a certain change in attitude; but this doesn’t alter the fact that group theory, finite and infinite, is saturated with combinatorics. The editors of group theorist Roger Lyndon’s selected works wrote:
“Lyndon produces elegant mathematics and thinks in terms of broad and deep ideas … I once asked him whether there was a common thread to the diverse work in so many different fields of mathematics, he replied that he felt the problems on which he had worked had all been combinatorial in nature.”
One of my best recent papers (with Rosemary Bailey, Cheryl Praeger and Csaba Schneider) showed that, in a certain higher-dimensional analogue of a Latin square, a group is encoded within the structure, which indeed represents the geometry of diagonal groups (one of the O’Nan–Scott types).
Another story I use a lot concerns the sporadic simple group found by Higman and Sims in a single evening, certainly the simplest construction of any of the sporadic simple groups. They were able to find it so easily because of their knowledge of relevant combinatorics (the Steiner systems of Witt).
And a final shot. One of the editor’s arguments was that fewer people would read a group theoretic paper in a discrete mathematics journal. I am not sure about that. I have recently observed that of all the papers by the outstanding group theorist Philip Hall, the one with the greatest number of citations by a huge margin is his paper titled “On representatives of subsets”, perhaps his one purely combinatorial paper. Not strictly comparable perhaps, since it was in a general journal; but a paper on combinatorics in a discrete mathematics journal is not likely to be starved of readers even if it has the word “group” in the title.
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