Obtaining Online Ecological Colourings by Generalizing First-Fit
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 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, and to show that deciding whether (G,c) is ecologically extendible can be done in polynomial time. We also describe when the extension is possible using only colours from a given finite setC.
graph colouring graph homomorphisms online algorithms