Last week I was at the 8th Slovenian Conference on Graph Theory.  This was the latest in what is commonly known as the ‘Bled conference’ but this year was in Kranjska Gora. This meant that the conference excursion was to Lake Bled. It was a very enjoyable conference with lots of interesting talks and it was good to catch up with lots of people. I was one of the plenary speakers and my talk was entitled ‘Bounding the number of automorphisms of a graph’. This surveyed the recent work on the Weiss conjecture and its generalisation the PSV conjecture. It also discussed my recent work with Luke Morgan on the PSV conjecture for semiprimitive groups with a nilpotent regular normal subgroup.  More details can be found on my slides.

Last week I was at the Symmetries of Graphs and Networks IV conference at Rogla in Slovenia. The conference webpage is here.  At the same time was the annual  PhD summer school in discrete maths. As usual it was a very enjoyable and well organised conference. It was good to catch up with some of the regulars and meet a few new people as well

I was one of the invited speakers and spoke about some of the work that I have been doing recently with Luke Morgan on graph-restrictive permutation groups. The slides are available here.  The two relevant preprints are on the arxiv here and here.

Now that I am back in Perth from the 33rd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, I can give a summary report of the conference.

Overall the conference was a great success. On the mathematical side there were many interesting talks with all the other plenary talks being good. A brief summary is as follows:

The second plenary talk on Monday was by Barbara Maenhaut from The University of Queensland who spoke on `Graph decompositions and factorisations’. This surveyed recent results in this area.  One was the conjecture that every complete graph of even order has a factorisation into 1-factors such that the union of any two is a Hamiltonian cycle. Such factorisations are called perfect 1-factorisations.

Both plenary talks on Tuesday mentioned the work of our own Gordon Royle. The first was by Adrian Bondy from the Institut Camille Jordan,  who spoke on ‘Aspects of Reconstruction’. The original reconstruction conjecture is that every graph on at least three vertices can be recognised from the set of subgraphs formed from it by deleting a vertex.   Adrian spoke about recent work on a variation of the problem where for each vertex you complement the sets of incident edges. It is conjectured that any graph on at least 4 vertices in switching reconstuctible.  Gordon got a mention, as along with Mark Ellingham, he has proved that this is true for triangle free graphs.  There is also a version in the digraph case where for each vertex v you take the graph formed by switching the direction of the arcs incident with v.

The second was by Luis  Goddyn from Simon Fraser University who spoke on `Excluded minors for bicirculant matroids’.  A circular matroid of a graph has elements the edges of the graph and the minimal dependent sets are the cycles.  A bicircular matroid for a graph has minimal dependent sets the connected subgraphs consisting of two cycles.  They currently have a list of 27 excluded minors and believe this is complete but have a small number of cases of possible ranks and number of elements which need to be checked. This used Gordon’s catalogue with Dillon Mayhew for small matroids.  The talk was also notable for Luis’s interesting dress sense.

Jaroslav Nesetril from Charles University spoke on `Existence vs Counting (on large and sparse structures).’ The main idea was to take a family of graphs and then form a new family of graphs consisting of all graphs in the original family plus those obtained by either contracting an edge or deleting a vertex. This process can be repeated. If you eventually obtain all graphs the original class is call somewhere dense while if you don”t it is called nowhere dense. For example, the set of planar graphs is nowhere dense while the set of all graphs of valency at most 3 is somewhere dense.

The final plenary talk was by Edy Baskoro from the Institut Teknologi Bandung, Indonesia,  who spoke `On the existence of almost moore digraphs.’ The maximal number of vertices of a graph of diameter d and valency k is at most $1+k+k^2+\ldots+k^d$.  A graph meeting this bound is called a Moore graph (see this earlier post),  while a graph whose number of vertices is 1 less than this bound is called an almost Moore graph. Similar questions can be asked in the directed case.

On the social side of things, the conference excursion was a trip to the Tamburlaine winery where we did some wine tasting and had a tour of the winery. We then went to another winery for lunch where we had the best conference excursion lunch I have ever had. The conference dinner was at Customs House where we were well fed and there was even a band. The best student talk prize was shared by Joanne Hall from RMIT and Beata Faller from the University of Canterbury.

I am back after my trip to the annual meeting of the Australian Maths Society in Adelaide. There were 12 plenary speakers this year. Going to one of these general meetings always reveals how big a subject mathematics is and how little of it I know. For this reason I always prefer “big picture” plenaries over more detailed ones. A couple of personal  highlights were Terry Tao’s public lecture on `Structure and randomness in the prime numbers’,  Jacqui Ramagge’s talk on `Totally disconnected, locally compact groups’ which outlined the theory of the subject developed mainly by George Willis,  and Akshay Venkatesh’s talk on his work with Ellenberg and Westerland on `The Cohen-Lenstra heuristics over global fields’.

There were up to 12 parallel sessions on at any given time. I mainly attended the combinatorics session where there were many good talks and  I spoke on `3/2-transitive permutation groups’. I have uploaded the slides for my talk. I  also ventured to the Topological groups session  and enjoyed Daniel Horadam’s talk there on `Automorphism groups of trees’. There were many other talks I would have liked to have gone to but they clashed with other interesting talks or me speaking.

Overall the conference went very well and the organisers should be congraulated on the great job they did. The conference dinner was also very well done: the food and wine was great and there was even a band.