Breaking the 2nBarrier for 5-Coloring and 6-Coloring

Or Zamir

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

    Abstract

    The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O∗(2n) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k "3, 4, better algorithms are known for the k-coloring problem. 3-coloring can be solved in O(1.33n) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73n) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k a 4 no improvements over the general O*(2n) are known. We show that both 5-coloring and 6-coloring can also be solved in O ((2-∈) n) time for some ∈ > 0. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ, α > 0, the chromatic number of graphs with at least α ·n vertices of degree at most Δ can be computed in O ((2 - ∈) n) time, for some ∈ = ∈Δ,α > 0. This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajlin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case of bounded-degree graphs.

    Original languageEnglish
    Title of host publication48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
    EditorsNikhil Bansal, Emanuela Merelli, James Worrell
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    ISBN (Electronic)9783959771955
    DOIs
    StatePublished - 1 Jul 2021
    Event48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 - Virtual, Glasgow, United Kingdom
    Duration: 12 Jul 202116 Jul 2021

    Publication series

    NameLeibniz International Proceedings in Informatics, LIPIcs
    Volume198
    ISSN (Print)1868-8969

    Conference

    Conference48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
    Country/TerritoryUnited Kingdom
    CityVirtual, Glasgow
    Period12/07/2116/07/21

    Keywords

    • Algorithms
    • Graph algorithms
    • Graph coloring

    Fingerprint

    Dive into the research topics of 'Breaking the 2nBarrier for 5-Coloring and 6-Coloring'. Together they form a unique fingerprint.

    Cite this