Let p and q be positive integers with p/q ≥ 2. The "reconfiguration problem" for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.

We prove a conjecture of Ohba that says that every graph G

The r-neighbour bootstrap percolation process on a graph G starts with an initial set A 0 of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A 0 percolates.We prove a conjecture of Balogh and Bollobás which says that, for fixed r and d → ∞, every percolating set in the d-dimensional hypercube has cardinality at least 1+o(1) r d r−1 . We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d-dimensional hypercube is d(d+3) 6 + 1 for all d ≥ 3.

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak ε-regular partition with the number of parts bounded by a polynomial in ε −1 . We construct a finitely forcible graphon W such that the number of parts in any weak ε-regular partition of W is at least exponential in ε −2 /2 5 log * ε −2 . This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.