 
Summary: DISCRETE
MATHEMATICS
ELSEVIER Discrete Mathematics 179 (1998) 111
The existence of uniquely G colourable graphs
D. Achlioptas a, J.I. Brown b'*, D.G. Corneil a, M.S.O. Molloy a
aDepartment of Computer Science, University of Toronto, Toronto, Canada
bDepartment of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax,
Nova Scotia, Canada B3H 3J5
Received 17 November 1995;received in revised form 31 October 1996;accepted 11 November 1996
Abstract
Given graphs F and G and a nonnegative integer k, a function n : V(F) ~ {1..... k} is a G
kcolouring of F if no induced copy of G is monochromatic; F is G kchromatic if F has a
G kcolouring but no G (k  1)colouring. Further, we say F is uniquely G kcolourable
if F is G kchromatic and, up to a permutation of colours, it has only one G kcolouring.
Such notions are extensions of the wellknown 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 kcolourable graphs, We prove the conjecture and show that, in fact, in all
cases infinitely many such graphs exist.
1. Introduction
There have been many generalizations of the notion of a vertex colouring of a
