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Article Dans Une Revue Combinatorics, Probability and Computing Année : 2022

Clustered 3-Colouring Graphs of Bounded Degree

Vida Dujmović
  • Fonction : Auteur
Louis Esperet
Pat Morin
  • Fonction : Auteur
Bartosz Walczak
  • Fonction : Auteur
David R. Wood
  • Fonction : Auteur

Résumé

A (not necessarily proper) vertex colouring of a graph has "clustering" $c$ if every monochromatic component has at most $c$ vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

Dates et versions

hal-02975327 , version 1 (22-10-2020)

Identifiants

Citer

Vida Dujmović, Louis Esperet, Pat Morin, Bartosz Walczak, David R. Wood. Clustered 3-Colouring Graphs of Bounded Degree. Combinatorics, Probability and Computing, 2022, 31 (1), pp.123-135. ⟨10.1017/S0963548321000213⟩. ⟨hal-02975327⟩
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