2020/07/21 by Or Zamir · 1 citation
Computer Science · Engineering · Mathematics · #1-planar graph #Advanced Graph Theory Research #Bounded function #Brooks' theorem #Chordal graph #Chromatic scale #Combinatorics #Complete coloring #Computer science #Degree (music) #Discrete mathematics #Edge coloring #Fractional coloring #Graph #Graph Labeling and Dimension Problems #Graph coloring #Graph power #Greedy coloring #Line graph #List coloring #Mathematics #Physics #Theoretical computer science #graph theory and CDMA systems
paper · pdf · doi:10.4230/lipics.icalp.2021.113
openalex publication_date 2021/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 > 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-ε) ⁿ) time for some ε > 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-ε) ⁿ) time, for some ε = εΔ,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.