Obtaining online ecological colourings by generalizing first-fit

Abstract

A colouring of a graph is ecological if every pair of vertices that have the same set of colours in their neighbourhood are coloured alike. We consider the following problem: if a graph G and an ecological colouring c of G are given, can further vertices added to G, one at a time, be coloured using colours from some finite set C so that at each stage the current graph is ecologically coloured? If the answer is yes, then we say that the pair (G,c) is ecologically online extendible. By generalizing the well-known First-Fit algorithm, we are able to characterize when (G,c) is ecologically online extendible. For the case where c is a colouring of G in which each vertex is coloured distinctly, we give a simple characterization of when (G,c) is ecologically online extendible using only the colours of c, and we also show that (G,c) is always online extendible if we permit ourselves to use one extra colour. We also study (off-line) ecological H-colourings where the colouring must satisfy further restrictions imposed by some fixed pattern graph H. We characterize the computational complexity of this problem. This solves an open question posed by Crescenzi et al.