8 June 2021, 3pm AEST Gordon Royle (University of Western Australia) |

Real chromatic roots of planar graphs |

(Part of the Round the World Relay in Combinatorics!) |

Abstract:
The chromatic polynomial P(G,q) of a graph G is the function that, for positive integer q, counts the number of proper q-colourings of the graph. The resulting function is a polynomial so, whether or not it makes any combinatorial sense, we can evaluate it at any complex number and therefore find its roots which may be integer, real of complex. The overall aim of this heavily studied topic is to understand the relationship between the properties of a graph and the location of its chromatic roots.
In this talk, I will focus on the real roots of the chromatic polynomials of planar graphs. I will start by giving some of the history, background and basic results, before focussing on efforts (by myself and others) to show that planar chromatic roots are dense in the real interval (3,4). The talk is mostly at a general level and anyone familiar with the basic concepts of graphs, proper colourings of graphs and polynomials will be able to follow the gist of it. |

28 April 2021, 5pm AEST Abhishek Methuku (University of Birmingham) |

A proof of the Erdős–Faber–Lovász conjecture |

Abstract:
The celebrated Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will sketch a proof of this conjecture for every large n.
A brief history of the problem can be found here. Joint work with D. Kang, T. Kelly, D. Kühn and D. Osthus. |