ON UNIQUELY -G k-COLOURABLE GRAPHS

Abstract

Given graphs F and G and a nonnegative integer k, a map Π: V(F) → + {lm …, k} is a -G k-colouring of F if the subgraphs induced by each colour class do not contain G as an induced subgraph; 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 and only 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. In a previous paper (J. Graph. Th. 11 (1987), 87–99), the authors conjectured that for all graphs G of order at least two and all nonnegative integers k there exist uniquely -G k-colourable graphs. We show here that the conjecture holds whenever G or its complement is 2-connected.