Improved bounds for centered colorings
Abstract
A vertex coloring $\phi$ of a graph $G$ is $p$centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$centered colorings with $\mathcal{O}(p^3\log p)$ colors where the previous bound was $\mathcal{O}(p^{19})$; (2) bounded degree graphs admit $p$centered colorings with $\mathcal{O}(p)$ colors while it was conjectured that they may require exponential number of colors in $p$; (3) graphs avoiding a fixed graph as a topological minor admit $p$centered colorings with a polynomial in $p$ number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no nontrivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$centered coloring and this bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$centered coloring.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 DOI:
 10.48550/arXiv.1907.04586
 arXiv:
 arXiv:1907.04586
 Bibcode:
 2019arXiv190704586D
 Keywords:

 Mathematics  Combinatorics;
 05XX;
 68Wx
 EPrint:
 Advances in Combinatorics, 2021:8