Complexity of Grundy Coloring and Its Variants

Abstract

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We show that Grundy Coloring can be solved in time $O^*(2.443^n)$ on graphs of order n. While the problem is known to be solvable in time $f(k,w)\cdot n$ for graphs of treewidth w, we prove that under the Exponential Time Hypothesis, it cannot be computed in time $O^*(c^{w})$, for any constant c. We also study the parameterized complexity of Grundy Coloring parameterized by the number of colors, showing that it is in $\mathsf {FPT}$ for graphs including chordal graphs, claw-free graphs, and graphs excluding a fixed minor.

Finally, we consider two previously studied variants of Grundy Coloring, namely Weak Grundy Coloring and Connected Grundy Coloring. We show that Weak Grundy Coloring is fixed-parameter tractable with respect to the weak Grundy number. In stark contrast, it turns out that checking whether a given graph has connected Grundy number at least k is $\mathsf {NP}$-complete already for $k=7$.

F. Foucaud—This research was done while this author was a postdoctoral fellow at the Department of Mathematics of University of Johannesburg (South Africa) and at LAMSADE.