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# Grundy Coloring & Friends, Half-Graphs, Bicliques

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LIPIcs.STACS.2020.58.pdf
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## Acknowledgements

A part of this work was done while the authors attended the "2019 IBS Summer research program on Algorithms and Complexity in Discrete Structures", hosted by the IBS discrete mathematics group. The third and the fourth authors partially supported by the ANR project “ESIGMA” (ANR-17-CE23-0010).

## Cite As

Pierre Aboulker, Édouard Bonnet, Eun Jung Kim, and Florian Sikora. Grundy Coloring & Friends, Half-Graphs, Bicliques. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.58

## Abstract

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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Fixed parameter tractability
##### Keywords
• Grundy coloring
• parameterized complexity
• ETH lower bounds
• K_{t,t}-free graphs
• half-graphs

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