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MATHEMATICS ELSEVIER Discrete Mathematics 179 (1998) 1-11
 

Summary: DISCRETE
MATHEMATICS
ELSEVIER Discrete Mathematics 179 (1998) 1-11
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
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.
1. Introduction
There have been many generalizations of the notion of a vertex colouring of a

  

Source: Achlioptas, Dimitris - Department of Computer Engineering, University of California at Santa Cruz

 

Collections: Computer Technologies and Information Sciences