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

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

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

All Science Journal Classification (ASJC) codes

  • Software

Fingerprint

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

Cite this