Islands in minor-closed classes. I. Bounded treewidth and separators
Combinatorics
2017-10-10 v1
Abstract
The clustered chromatic number of a graph class is the minimum integer such that for some the vertices of every graph in the class can be colored in colors so that every monochromatic component has size at most . We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve some extremal problems on bootstrap percolation of minor-closed classes.
Cite
@article{arxiv.1710.02727,
title = {Islands in minor-closed classes. I. Bounded treewidth and separators},
author = {Zdeněk Dvořák and Sergey Norin},
journal= {arXiv preprint arXiv:1710.02727},
year = {2017}
}