The existence of uniquely − G colourable graphs

Abstract

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.