The good, the bad, and the great: Homomorphisms and cores of random graphs

We consider homomorphism properties of a random graph G(n, p), where p is a function of n. A core H is great if for all e ∈ E(H), there is some homomorphism from H−e to H that is not onto. Great cores arise in the study of uniquely H-colourable graphs, where two inequivalent definitions arise for general cores H. For a large range of p, we prove that with probability tending to 1 as n → ∞, G ∈ G(n, p) is a core that is not great. Further, we give a construction of infinitely many non-great cores where the two definitions of uniquely H-colourable coincide.