Grundy Coloring and Friends, Half-Graphs, Bicliques - Aboulker, Pierre; Bonnet, Édouard; Kim, Eun Jung; Sikora, Florian (Algorithmica, 2023)
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f ( k ) n 2 k - 1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b - Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS ’17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K t , t -free graphs for b - Chromatic Core and Partial Grundy Coloring , making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.