The existence of uniquely -G colourable graphs

Given graphs F and G and a nonnegative integer k, a function n : V(F) ~ {1 . . . . . k} is a G k-colouring of F if no induced copy of G 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.