3-Colouring Planar Graphs
Vida Dujmovi\'c, Pat Morin, Sergey Norin, David R. Wood

TL;DR
This paper proves that every planar graph with n vertices can be colored with three colors such that each monochromatic component has size proportional to n^{4/9}, improving previous bounds.
Contribution
It establishes a new bound on monochromatic component size in 3-colorings of planar graphs, advancing understanding of graph coloring properties.
Findings
Monochromatic components are of size O(n^{4/9})
Improves previous bound of O(n^{1/2})
Advances theoretical understanding of planar graph colorings
Abstract
We show that every -vertex planar graph is 3-colourable with monochromatic components of size . The best previous bound was due to Linial, Matou\v{s}ek, Sheffet and Tardos [Combin. Probab. Comput., 2008].
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