Summary
Given an approximate vertex coloring algorithm, a graph is said to be slightly hard-to-color if some implementation of the algorithm uses more colors than the minimum needed. Similarly, a graph is said to be hard-to-color if every implementation of the algorithm results in a non-optimal coloring. So far the smallest hard-to-color and/or slightly hard-to-color graphs have been known only for the following algorithms: greedy independent sets, random sequential, connected sequential, largest-first and smallest-last. In the paper we give such graphs for the following heuristics: saturation largest-first, largest-first with interchange, smallest-last with interchange and coloring pairs. Most of the new results have been obtained by means of an exhaustive computational search.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Babel L., TinhoferG., Hard-to-color graphs for connected sequential colorings, Tech. Rep. TUM-M9104, Tech. Univ. Miinchen, 1991 (to appear in Ann. Disc. Math).
Brelaz D., New methods to color the vertices of a graph, Comm. ACM 22, 1979, 251–256.
Hansen P., Kuplinsky J., Slightly hard-to-color graphs, Congr Numer. 78, 1990, 81–98.
Hansen P., Kuplinsky J., The smallest hard-to-color graph, Disc. Math. 96, 1991, 199–212.
Johnson D.S., Approximation algorithms for combinatorial problems, J. Comp. Syst. Sci. 9, 1974, 256–278.
Johnson D.S., Worst-case behavior of graph coloring algorithms, Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Utilitas Mathematicae, Winnipeg, 1974, 513–527.
Kubale M., Pakulski J., Piwakowski K., The smallest hard-to-color graph for the SL algorithm (presented at 2nd Kraków Conf. on Graph Theory, Zakopane, 1994 ).
Matula D.W., Marble G., Isaacson D.: Graph coloring algorithms, in: Graph Theory and Computing, Academic Press, New York, 1972, 109–122.
Welsh D. J., Powell M.B., An upper bound for the chromatic number of a graph and its application to timetabling problem, Comp. J. 10, 1967, 85–86.
Wood D.C., A technique for coloring a graph applicable to large scale timetabling problems, Comp. J. 12, 1969, 317–319.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kubale, M., Pakulski, J. (1995). A Catalogue of the Smallest Hard-to-Color Graphs. In: Derigs, U., Bachem, A., Drexl, A. (eds) Operations Research Proceedings 1994. Operations Research Proceedings, vol 1994. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79459-9_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-79459-9_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58793-4
Online ISBN: 978-3-642-79459-9
eBook Packages: Springer Book Archive