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.