Obtaining online ecological colourings by generalizing first-fit

Johnson, M.; Patel, V.; Paulusma, D.; Trunck, T.

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.

Citation

Johnson, M., Patel, V., Paulusma, D., & Trunck, T. (2010, December). Obtaining online ecological colourings by generalizing first-fit. Presented at 5th International Computer Science Symposium in Russia (CSR 2010), Kazan, Russia

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 5th International Computer Science Symposium in Russia (CSR 2010)
• Publication Date: Jun 20, 2010
• Deposit Date: Oct 6, 2010
• Publicly Available Date: Oct 29, 2010
• Print ISSN: 0302-9743
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 6072
• Pages: 240-251
• Series Title: Lecture notes in computer science
• Series Number: 6072
• Series ISSN: 0302-9743,1611-3349
• Edition: 5th
• Book Title: Computer science : theory and applications : 5th International Computer Science Symposium in Russia, CSR, 16-20 June 2010, Kazan, Russia ; proceedings.
• ISBN: 9783642131813
• DOI: https://doi.org/10.1007/978-3-642-13182-0_22
• Public URL: https://durham-repository.worktribe.com/output/1697609
• Additional Information: Computer Science – Theory and Applications : 5th International Computer Science Symposium in Russia, CSR, 16-20 June 2010, Kazan, Russia ; Proceedings.