The honeycomb toroidal graphs are a family of graphs I have been looking at now and then for thirty years. I shall discuss an ongoing project dealing with hamiltonicity as well as some of their properties which have recently interested the computer architecture community.

Finite generalised polygons are the rank 2 irreducible spherical buildings, and include projective planes and the generalised quadrangles, hexagons, and octagons. Since the early work of Ostrom and Wagner on the automorphism groups of finite projective planes, there has been great interest in what the automorphism groups of generalised polygons can be, and in particular, whether it is possible to classify generalised polygons with a prescribed symmetry condition. For example, the finite Moufang polygons are the 'classical' examples by a theorem of Fong and Seitz (1973-1974) (and the infinite examples were classified in the work of Tits and Weiss (2002)). In this talk, we give an overview of some recent results on the study of symmetric finite generalised polygons, and in particular, on the work of the speaker with Cai Heng Li and Eric Swartz.

In this talk I'll describe some recent discoveries about edge-transitive graphs and edge-transitive maps. These are objects that have received relatively little attention compared with their vertex-transitive and arc-transitive siblings.
First I will explain a new approach (taken in joint work with Gabriel Verret) to finding all edge-transitive graphs of small order, using single and double actions of transitive permutation groups. This has resulted in the determination of all edge-transitive graphs of order up to 47 (the best possible just now, because the transitive groups of degree 48 are not known), and bipartite edge-transitive graphs of order up to 63. It also led us to the answer to a 1967 question by Folkman about the valency-to-order ratio for regular graphs that are edge- but not vertex-transitive.
Then I'll describe some recent work on edge-transitive maps, helped along by workshops at Oaxaca and Banff in 2017. I'll explain how such maps fall into 14 natural classes (two of which are the classes of regular and chiral maps), and how graphs in each class may be constructed and analysed. This will include the answers to some 18-year-old questions by Širáň, Tucker and Watkins about the existence of particular kinds of such maps on orientable and non-orientable surfaces.

I am going to look at three unsolved graph theory problems for which the same family of graphs presents a barrier to either solving or making substantial progress on the problems. The graphs in this family are called honeycomb toroidal graphs. The three problems are not closely related.

Targeted Audience: All early career staff and PhD students; other staff welcome

Abstract: Many of us have been involved in discussions revolving around the problem of choosing suitable thesis topics and projects for post-graduate students, honours students and vacation research students. The panel is going to present some ideas that we hope people in the audience will find useful as they get ready for or continue with their careers.

About the Speakers: Professor Brian Alspach has supervised thirteen PhDs, twenty-five MScs, nine post-doctoral fellows and a dozen undergraduate scholars over his fifty-year career. Professor Eric Beh has 20 years' international experience in the analysis of categorical data with a focus on data visualisation. He has and has, or currently is, supervised about a 10 PhD students. Dr Mike Meylan has twenty years research experience in applied mathematics both leading projects and working with others. He has supervised 5 PhD students and three post-doctoral fellows.

Today's discrete mathematics seminar is dedicated to Mirka Miller. I am going to present the beautiful Hoffman-Singleton (1964) paper which established the possible values for valencies for Moore graphs of diameter 2, gave us the Hoffman-Singleton graph of order 50, and gave us one of the intriguing still unsettled problems in combinatorics. The proof is completely linear algebra and is a proof that any serious student in discrete mathematics should see sometime. This is the general area in which Mirka made many contributions.

This afternoon (31 October) we shall complete the discussion about vertex-minimal graphs with dihedral automorphism groups. I have attached an outline of what was covered in the first two weeks.

This week we shall continue by introducing the cast of characters to be used for producing minimal-order graphs with dihedral automorphism group.

Konig (1936) asked whether every finite group G is realized as the automorphism group of a graph. Frucht answered the question in the affirmative and his answer involved graphs whose orders were substantially bigger than the orders of the groups leading to the question of finding the smallest graph with a fixed automorphism group. We shall discuss some of the early work on this problem and some recent results for the family of dihedral groups.

This week I shall finish my discussion of sequenceable and R-sequenceable groups.

I am now refereeing a manuscript on the above and I’ll tell you about its contents.

B. Gordon (1961) defined sequenceable groups and G. Ringel (1974) defined R-sequenceable groups. Friedlander, Gordon and Miller conjectured that finite abelian groups are either sequenceable or R-sequenceable. The preceding definitions are special cases of what T. Kalinowski and I are calling an orthogonalizeable group, namely, a group for which every Cayley digraph on the group admits either an orthogonal directed path or an orthogonal directed cycle. I shall go over the history and current status of this topic along with a discussion about the completion of a proof of the FGM conjecture.

Mathematicians sometimes speak of the beauty of mathematics which to us is reflected in proofs and solutions for the most part. I am going to give a few proofs that I find very nice. This is stuff that post-grad discrete students certainly should know exists.

This week I shall conclude my discussion of pancyclicity and Cayley graphs on generalized dihedral groups.

This is joint work with our former honours student Alex Muir. We look at the variety of lengths of cycles in Cayley graphs on generalized dihedral groups.

This is joint work with our former honours student Alex Muir. We look at the variety of lengths of cycles in Cayley graphs on generalized dihedral groups.

This week I shall finish my discussion about searching graphs by looking at the recent paper by Clarke and MacGillavray that characterizes graphs that are k-searchable.

This Thursday, sees a return to graph searching in the discrete mathematics instructional seminar. I’ll be looking at characterization results.

Brian Alspach will continue discussing searching graphs embedded on the torus.

This week I shall continue the discussion of searching graphs.

This week I shall start a series of talks on basic pursuit-evasion in graphs (frequently called cops and robber in the literature). We shall do some topological graph theory leading to an intriguing conjecture, and we'll look at a characterization problem.

I shall be describing a largely unexplored concept in graph theory which is, I believe, an ideal thesis topic. I shall be presenting this at the CIMPA workshop in Laos in December.

We shall finish our look at two-sided group graphs.

What do the three elements of the title have in common is the utility of using graph searching as a model. In this talk I shall discuss the relatively brief history of graph searching, several models currently being employed, several significant results, unsolved conjectures, and the vast expanse of unexplored territory.

I am refereeing a manuscript in which a new construction for producing graphs from a group is given. There are some surprising aspects of this new method and that is what I shall discuss.

This year is the fiftieth anniversary of Ringel's posing of the well-known graph decomposition problem called the Oberwolfach problem. In this series of talks, I shall examine what has been accomplished so far, take a look at current work, and suggest a possible new avenue of approach. The material to be presented essentially will be self-contained.

This year is the fiftieth anniversary of Ringel's posing of the well-known graph decomposition problem called the Oberwolfach problem. In this series of talks, I shall examine what has been accomplished so far, take a look at current work, and suggest a possible new avenue of approach. The material to be presented essentially will be self-contained.

This year is the fiftieth anniversary of Ringel's posing of the well-known graph decomposition problem called the Oberwolfach problem. In this series of talks, I shall examine what has been accomplished so far, take a look at current work, and suggest a possible new avenue of approach. The material to be presented essentially will be self-contained.

: In this final talk of the sequence we will sketch Blinovsky's recent proof of the conjecture: Whenever n is at least 4k, and A is a set of n numbers with sum 0, then there are at least (n-1) choose (k-1) subsets of size k which have non-negative sum. The nice aspect of the proof is the combination of hypergraph concepts with convex geometry arguments and a Berry-Esseen inequality for approximating the hypergeometric distribution. The not so nice aspect (which will be omitted in the talk) is the amount of very tedious algebraic manipulation that is necessary to verify the required estimates. There are slides for all four MMS talks here.

The Erdos-Ko-Rado (EKR) Theorem is a classical result in combinatorial set theory and is absolutely fundamental to the development of extremal set theory. It answers the following question: What is the maximum size of a family F of k-element subsets of the set {1,2,...,n} such that any two sets in F have nonempty intersection?

In the 1980's Manickam, Miklos and Singhi (MMS) asked the following question: Given that a set A of n real numbers has sum zero, what is the smallest possible number of k-element subsets of A with nonnegative sum? They conjectured that the optimal solutions for this problem look precisely like the extremal families in the EKR theorem. This problem has been open for almost 30 years and many partial results have been proved. There was a burst of activity in 2012, culminating in a proof of the conjecture in October 2013.

This series of talks will explore the basic EKR theorem and discuss some of the recent results on the MMS conjecture.

The Erdos-Ko-Rado (EKR) Theorem is a classical result in combinatorial set theory and is absolutely fundamental to the development of extremal set theory. It answers the following question: What is the maximum size of a family F of k-element subsets of the set {1,2,...,n} such that any two sets in F have nonempty intersection?

In the 1980's Manickam, Miklos and Singhi (MMS) asked the following question: Given that a set A of n real numbers has sum zero, what is the smallest possible number of k-element subsets of A with nonnegative sum? They conjectured that the optimal solutions for this problem look precisely like the extremal families in the EKR theorem. This problem has been open for almost 30 years and many partial results have been proved. There was a burst of activity in 2012, culminating in a proof of the conjecture in October 2013.

This series of talks will explore the basic EKR theorem and discuss some of the recent results on the MMS conjecture.

The Erdos-Ko-Rado (EKR) Theorem is a classical result in combinatorial set theory and is absolutely fundamental to the development of extremal set theory. It answers the following question: What is the maximum size of a family F of k-element subsets of the set {1,2,...,n} such that any two sets in F have nonempty intersection?

In the 1980's Manickam, Miklos and Singhi (MMS) asked the following question: Given that a set A of n real numbers has sum zero, what is the smallest possible number of k-element subsets of A with nonnegative sum? They conjectured that the optimal solutions for this problem look precisely like the extremal families in the EKR theorem. This problem has been open for almost 30 years and many partial results have been proved. There was a burst of activity in 2012, culminating in a proof of the conjecture in October 2013.

This series of talks will explore the basic EKR theorem and discuss some of the recent results on the MMS conjecture.

In a recent referee report, the referee said he/she could not understand the proofs of either of the two main results. Come and judge for yourself! This is joint work with Darryn Bryant and Don Kreher.

In 1966 Gallai conjectured that a connected graph of order n can be decomposed into n/2 or fewer paths when n is even, or (n+1)/2 or fewer paths when n is odd. We shall discuss old and new work on this as yet unsolved conjecture.

In 1966 Gallai conjectured that a connected graph of order n can be decomposed into n/2 or fewer paths when n is even, or (n+1)/2 or fewer paths when n is odd. We shall discuss old and new work on this as yet unsolved conjecture.

This week Brian Alspach will complete the discussion on Burnside's Theorem and vertex-transitive graphs of prime order.

We shall continue exploring implications of Burnside's Theorem for vertex-transitive graphs.

We shall continue exploring implications of Burnside's Theorem for vertex-transitive graphs.

Burnside's Theorem characterising transitive permutation groups of prime degree has some wonderful applications for graphs. This week we start an exploration of this topic.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

This week Brian Alspach concludes his series of talks entitled "The Anatomy Of A Famous Conjecture." We shall be in V27 - note room change.

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

Brian Alspach will continue with "The Anatomy Of A Famous Conjecture" this Thursday. One can easily pick up the thread this week without having attended last week, but if you miss this week it will not be easy to join in next week.

I have embarked on a project of looking for Hamilton paths in Cayley graphs on finite Coxeter groups. This talk is a report on the progress thus far.

Brian Alspach will continue with "The Anatomy of a Famous Conjecture" this Thursday. One can easily pick up the thread this week without having attended last week, but if you miss this week it will not be easy to join in next week.

In my opinion, the most significant unsolved problem in graph decompositions is the cycle double conjecture. This begins a series of talks on this conjecture in terms of background, relations to other problems and partial results.

This week we shall conclude our look at the paper by Dave Witte on Hamilton directed cycles in Cayley digraphs of prime power order.

We shall start looking at Dave Witte's (now Dave Morris) proof that connected Cayley digraphs of prime power order have Hamilton directed cycles.

This week the discrete mathematics instructional seminar will continue with a consideration of the Lovasz problem. This is the last meeting of the seminar until 13 October.

The topic is Lovasz's Conjecture that all connected vertex-transitive graphs have Hamilton paths.

This Thursday is your chance to start anew! I shall be starting a presentation of the best work that has been done on Lovasz's famous 1979 problem (now a conjecture) stating that every connected vertex-transitive graph has a Hamilton path. This is good stuff and requires minimal background.

This week we shall conclude the proof of the uniqueness of the Hoffman-Singleton graph.

We continue looking at the 1960 Hoffman-Singleton paper about Moore graphs and related topics.

We meet this Thursday at the usual time when I will show you a nice application of the Edmonds-Fulkerson matroid partition theorem, namely, I'll prove that Paley graphs have Hamilton decompositions (an unpublished result).

This week we shall start the classical paper by Jack Edmonds and DR Fulkerson on partitioning matroids.

We shall conclude the discussion of some of the mathematics surrounding Birkhoff's Theorem about doubly stochastic matrices.

This is a discrete mathematics instructional seminar commencing 24 February--to meet on subsequent Thursdays from 3:00-4:00 p.m. The seminar will focus on "classical" papers and portions of books.