The existence of uniquely -G colourable graphs

The author can archive pre-print, post-print of the article. appropriate journal homepage link is attached.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 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