The smallest hard-to-color graph for the SL algorithm Original Research Article

For a given approximate vertex coloring algorithm a graph is said to be slightly hard-to-color (SHC) if some implementation of the algorithm uses more colors than the minimum needed. Similarly, a graph is said to be hard-to-color (HC) if every implementation of the algorithm results in a nonoptimal coloring. We study smallest such graphs for the smallest-last (SL) coloring algorithm. Our main result is that the prism is a unique smallest SHC graph and the prismatoid is a unique smallest HC graph for the SL algorithm.