PC Seminar: The threshold for rainbow F-factors and a spread condition for colourings

Abstract: We establish the threshold for a randomly coloured random graph to contain a rainbow copy of an F-factor (where F is any fixed 1-balanced graph), answering a question of Han and Yuan. Along the way we prove a theorem generalising recent work of Han and Yuan, where instead of rainbow colourings we consider a more general class of colourings satisfying a natural spread condition. The proof also involves several careful couplings including a coupling technique of McDiarmid (twice) and a coupling of Riordan which translates between copies of F in G(n,p) and the r-uniform random hypergraph (where r is the number of vertices of F). Joint work with Annika Heckel.

This is a seminar in our seminar series on Probability and Combinatorics (PC).