Primitive groups synchronize non-uniform maps of extreme ranks

Let Ω be a set of cardinality n, G a permutation group on Ω, and f : Ω → Ω a map which is not a permutation. We say that G synchronizes f if the semigroup G, f contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree n primitive groups synchronize maps of rank n − 1 (thus, maps with kernel type (2, 1,..., 1)). We prove some extensions of Rystsov’s result, including this: a primitive group synchronizes every map whose kernel type is (k, 1,..., 1). Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and n − 2. These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undirected) graph. The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.