Random Perfect Graphs

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $\alpha(P_n)$ and clique number $\omega(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any given graph $H$ as an induced subgraph is asymptotically $0$ or $\frac12$ or $1$. Further we show that almost all perfect graphs are $2$-clique-colourable, improving a result of Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have connectivity $\kappa(P_n)$ equal to their minimum degree; they are almost all in class one (edge-colourable using $\Delta$ colours, where $\Delta$ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon $W_P(x, y) = \frac12(\mathbb{1}[x \le 1/2] + \mathbb{1}[y \le 1/2])$

Measurements of edge uncolourability in cubic graphs

Philosophiae Doctor - PhDThe history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance

Snarks

Abstract: An edge colouring of a graph is an assignment of labels (colours) to the edges of a graph such that adjacent edges are assigned different colours. It is clear that if the maximum degree of the graph is k, and v is a vertex of degree k, then at least k colours are needed to colour the edges incident with v. A famous result by the Russian mathematician Vadim Vizing states that k+1 colours will always suffice to colour the edges of the whole graph. Graphs whose edges can be coloured with its maximum degree number of colours are called Class one graphs and the rest are called Class two graphs. A snark is a 3-regular Class two graph that satisfies some additional requirements, depending on whose definition one follows. Certainly, they should be connected and bridgeless. Modern authors usually require that they be triangle-free, or even of girth at least 5, and cyclically 4-edge connected. They have been studied since the 1880’s, when the Scottish physicist Peter Tait proved that the Four Colour Theorem is equivalent to the statement that no snark is planar. The popular science writer Martin Gardner gave them the name “snark” in 1975. The name, taken from the elusive creature in Lewis Carroll’s poem The Hunting of the Snark, reflects the scarcity of examples in the years after Tait defined them. The smallest and earliest known example of a snark is the Petersen graph, discovered in 1898. Due to their connection with the Four Colour Theorem (Four Colour Conjecture, at the time), much attention was given to the pursuit of new examples of snarks (with the hope of finding a planar one, perhaps), but a second example was not discovered until 1946. Since then, more examples have been discovered, including infinite families. I will discuss early examples and infinite families of snarks and their connections to well-known results and conjectures in graph theory, highlighting contributions made by women on snarks (Amanda Chetwynd, Myriam Preissmann, sarah-marie belcastro, Carla Fiori, Beatrice Ruini (all contemporary)) and other topics in graph theory (Henda Swart, 1939 – 2016, whose legacy is the top quality researchers whom she inspired, Fan Chung, Penny Haxell, to mention but a few). About the speaker: Kieka Mynhardt was born in Cape Town and lived in South Africa until 2002, when she moved to Victoria, Canada. She obtained her PhD from the University of Johannesburg under the supervision of Izak Broere. She started her professional career at the University of Pretoria, moving to the University of South Africa, also in Pretoria, after two years. She now holds a professorship at the University of Victoria.Non UBCUnreviewedAuthor affiliation: University of VictoriaFacult