I’m in France at the moment, having spent the last week at a workshop on matroid computation at Henry Crapo’s house in the tiny village of La Vacquerie et Saint Martin de Castries, but now I’ve moved on to Paris for a week.

I had been intending to drop in to see Michel Las Vergnas, but it seems that I somehow missed the sad news that he died in January. Although I only met him once, a few years go in Barcelona, we had frequently communicated about an exasperating conjecture of his.

A quick bit of background: start with the vertex-edge incidence matrix of the graph, such that the column corresponding to the edge ${e=vw}$ has exactly two non-zero entries, both equal to one, in rows ${v}$ and ${w}$. Viewed as a matrix over ${GF(2)}$, the row space and null space of this matrix are called the cocycle space ${C}$ and the cycle space ${C^\perp}$ of the graph. The non-zero vectors in ${C}$, called the cocycles, are the (characteristic vectors of the) cut-sets of the graph, while the non-zero vectors in ${C^\perp}$, called the cycles, are the subgraphs in which every vertex has even degree. (Notice that this notion of cycle is considerably broader than the usual graph-theoretic notion.) A set of edges that is simultaneously a cycle and a cocycle is called a bicycle, and the bicycle space is the vector space ${C \cap C^\perp}$. (A coding theorist would call this the hull of the linear code ${C}$.)

If ${T(x,y)}$ is the Tutte polynomial of the graph, then it is well known that

$\displaystyle T(-1,-1) = \pm \ 2^{d(C\cap C^\perp)}$

where ${d(C\cap C^\perp)}$ is the dimension of the bicycle space.

Among many other things, Las Vergnas was interested in a variety of evaluations of the Tutte polynomial along the diagonal where ${y=x}$, and so he considered the polynomial ${D(x) = T(x,x)}$ and (for reasons unknown to me) its derivates ${D'}$, ${D''}$, ${D'''}$, etc. all evaluated at the point ${x=-1}$. While he could not find an interpretation for ${D'(-1)}$, ${D''(-1)}$, ${\ldots}$ he noticed that the sequence of values generated appeared to be divisible by unexpectedly high powers of ${2}$, with each differentiation reducing the exponent of ${2}$ by at most one. So he made the following conjecture:

If ${G}$ is a graph with bicycle dimension ${d}$, and ${D(x) = T(x,x)}$ is its diagonal Tutte polynomial, then  ${2^{d-k} \mid D^{(k)}(-1)}$ for all ${k < d}$.

For example, for the graph ${K_6}$, we have

$\displaystyle D(x) = x^{10}+5 x^9+15 x^8+41 x^7+88 x^6+172 x^5+300 x^4+390 x^3+236 x^2+48 x$

and so ${D(-1) = -16}$, ${D'(-1) = 80}$, ${D''(-1) = -220}$, ${D'''(-1) = 270}$, which are indeed divisible by ${16}$, ${8}$, ${4}$ and ${2}$ respectively.

However a quick computer search revealed that as it stands, the conjecture is false — the graph ${K_8}$ is a counterexample. But I was slightly intrigued by now, so I searched a bit harder and noticed that no matter how hard I tried, I could not find any planar graphs that did not have the property of the conjecture. So I emailed Michel and we revised the conjecture to

If ${G}$ is a planar graph with bicycle dimension ${d}$, and ${D(x) = T(x,x)}$ is its diagonal Tutte polynomial, then  ${2^{d-k} \mid D^{(k)}(-1)}$ for all ${k < d}$.

And that’s basically where it stands. Although it is not a particularly important statement in itself, if it is true, then surely there must be some more combinatorial information to be wrung from the Tutte polynomial if only we knew how to look at it in the right way!

In fact, I think that planar graphs is not the precisely correct class to be working with, but rather some class that includes planar graphs; something perhaps like $\Delta Y$ graphs.

One of the central and important concepts in projective geometry is the beautiful connection between 3-dimensional projective space and the Klein quadric. As is indicated by the title, this correspondence between these two geometries was named after the German mathematician Felix Klein, who studied it in his dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form (1868). The Klein Correspondence can be used to give geometric understanding for certain isomorphisms of low rank classical groups, and we will give an example of some of these in a follow-up post. In geometry, the Klein Correspondence can sometimes illuminate an obscure object in projective 3-space, for the Klein quadric is naturally embedded in a 5-dimensional projective space where there is an added richness to the geometry, and one has available other ways to distinguish certain configurations. For example, if you were to learn about linear complexes in 3-dimensional projective space, you might find one class known as the “parabolic congruences” as a somewhat messy object of pencils of lines centered on the points of a common line. However, under the Klein Correspondence, a parabolic congruence becomes a quadratic cone: one of the first 3-dimensional geometric objects we first encountered in school mathematics. Notice that I have not specified the field we are working over; it won’t matter!

The day started badly. I woke at 3am to a sick and miserable one-year-old, I got stuck in a traffic jam on the way to work, and by lunchtime I was feeling down about the usual whackacademic innuendo. But at 3pm, Eric Swartz knocked on my door. Eric has stunningly proved an outstanding conjecture on generalised quadrangles which I’ll report on when I have his blessing.

Rather than let this blog die off due to lack of posts, I thought I’d write about a small but annoying problem that I’ve been thinking about recently, but without success, and a natural conjecture that has arisen from it.

[Note: On some browsers there appears to be a sizing problem with the images WordPress uses for LaTeX formulas, and they appear ten times the right size. If this happens, reloading the webpage seems to fix it]

The problem arose in the context of binary matroids, but because a binary matroid is really just a set of points in a binary vector space, it can be phrased entirely as a linear algebra problem.

So let ${M}$ be a set of non-zero vectors in the vector space ${V = GF(2)^r}$ such that ${M}$ spans ${V}$ and, for reasons to be clarified later, no vector in ${M}$ is independent of the others. In matroid terminology, this just says that ${M}$ is a simple binary matroid of rank ${r}$ with no coloops.

Then define a  basis of ${M}$ to be a linearly independent subset of ${M}$ of rank ${r}$ (in other words, just a basis of ${V}$ and a circuit of ${M}$ to be a minimally dependent set of vectors, i.e. a set of vectors that is linearly dependent but any proper subset of which is linearly independent.

As an example, take ${M = PG(2,2)}$ (a.k.a the Fano plane) — this means to take all the non-zero vectors in ${GF(2)^3}$. This has 28 bases (7 choices for a first vector ${v}$, 6 for a second vector ${w}$ and then 4 for the third vector which cannot be ${v}$, ${w}$ or ${v+w}$ , then all divided by 6 because this counts each basis ${6}$ times). It has 7 circuits of size 3 (each being of the form ${v}$, ${w}$, ${v+w}$) and 7 circuits of size 4, being the complements of the circuits of size 3, for a total of 14 circuits.

Letting ${b(M)}$, ${c(M)}$ denote the numbers of bases and circuits of ${M}$ respectively, the question is about the ratio of these two numbers. More precisely,

Determine a lower bound, in terms of the rank ${r}$, for the ratio ${\frac{b(M)}{c(M)}}$?

Harald Helfgott, a friend of SymOmega, has only yesterday uploaded an article on the arxiv claiming a proof of the “Ternary Goldbach Conjecture”:

Every odd integer $n>5$ is the sum of three primes.

This is big news in number theory since it is one of the best results on the Goldbach Conjecture to date. The “Ternary GC” also had its origins in the letters of Goldbach and Euler in 1742, and is sometimes referred to as Goldbach’s weak conjecture, since if every even number greater than 4 is the sum of two primes, then by adding the number 3 to the two primes will result in every odd number greater than 5 being the sum of three primes.

I’ve just posted a paper entitled “The Merino-Welsh Conjecture holds for Series-Parallel Graphs” to the arxiv which proves precisely the assertion given in the paper’s title.  This paper resulted from Steve Noble’s visit here last year, where we worked quite hard on this conjecture, and in fact I blogged about the problem then. The conjecture is that the number of spanning trees of a graph is dominated either by the number of acyclic orientations or by the number of totally cyclic orientations.

Previous results on the Merino-Welsh conjecture had proved the result for either very sparse graphs (where the number of spanning trees is dominated by the number of acyclic orientations) or very dense graphs (where the number of spanning trees is dominated by the number of totally cyclic orientations) regardless of the graph’s structure. However for intermediate-sized graphs — in particular those with $m = 2(n-1)$ — the graph structure counts, and there is a subtle interplay between the numbers of the two types of orientations.

Series-parallel graphs are always a great test case, because there is a natural decomposition of a series-parallel graph into smaller series-parallel graphs that provides an ideal inductive framework for a vast range of applications. For us, it provides enough structure to be able to keep track of the numbers of spanning trees and both types of orientation. Not entirely precisely, but enough to identify certain series-parallel graphs that cannot form part of a minimal counterexample to the conjecture. Working systematically, it was possible to ultimately exhaust all the possibilities for graphs that might be part of a minimal counterexample, thereby showing that this hypothetical counterexample cannot actually exist.

On one hand, I rather like this paper because the approach to coping with the interaction between the parameters seems novel and interesting, but on the other hand it relies heavily on the very constrained construction steps allowed in building series-parallel graphs and so will be almost impossible to generalise to larger classes!

A position here in the School of Maths and Stats at UWA has just been advertised.  It is a full-time continuing position as either an Assistant or Associate Professor (Level B/C)  and the school is looking for ` a generalist mathematician with excellent skills in teaching and learning to develop and deliver a range of first year units to students from across the University.’  More details are available at the link above and queries should be directed to the Head of School, Andrew Bassom.