The existence of uniquely −G colourable graphs Original Research Article

Given graphs F and G and a nonnegative integer k, a function π : V(F) → 1, …, k is a −G k-colouring of F if no induced copy of is monochromatic; F is −G k-chromatic if F has a −G k-colouring but no −G (k − 1)-colouring. Further, we say F is uniquely −G k-colourable if F is −G k-chromatic and, up to a permutation of colours, it has only one −G k-colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely −G k-colourable graphs. We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist.