The smallest hard-to-color graph

Abstract For a given 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. We study smallest such graphs for the sequential coloring algorithm. Our main result is that, for the largest-first algorithm, there is a unique smallest hard-to-color graph.