Detailed Information on Publication Record

DĘBSKI, Michał Karol, Stefan FELSNER, Piotr MICEK and Felix SCHRÖDER. Improved bounds for centered colorings. Online. In Shuchi Chawla. Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms. Not specified: SIAM, 2020, p. 2212-2226. ISBN 978-1-61197-599-4. Available from: https://dx.doi.org/10.1137/1.9781611975994.136.

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TY  - JOUR
ID  - 1645722
AU  - Dębski, Michał Karol - Felsner, Stefan - Micek, Piotr - Schröder, Felix
PY  - 2020
TI  - Improved bounds for centered colorings
PB  - SIAM
CY  - Not specified
SN  - 9781611975994
KW  - centered coloring
KW  - bounded expansion
KW  - planar graph
KW  - entropy compression
UR  - https://epubs.siam.org/doi/abs/10.1137/1.9781611975994.136
L2  - https://epubs.siam.org/doi/abs/10.1137/1.9781611975994.136
N2  - A vertex coloring c of a graph G is p-centered if for every connected subgraph H of G either c 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 >= 1, 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 O(p^3 log p) colors where the previous bound was O(p^19); (2) bounded degree graphs admit p-centered colorings with 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 non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require (p+t choose 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. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.
ER  -

Basic information
Original name	Improved bounds for centered colorings
Authors	DĘBSKI, Michał Karol (616 Poland, belonging to the institution), Stefan FELSNER (276 Germany), Piotr MICEK (616 Poland) and Felix SCHRÖDER.
Edition	Not specified, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, p. 2212-2226, 15 pp. 2020.
Publisher	SIAM

Other information
Original language	English
Type of outcome	Proceedings paper
Field of Study	10201 Computer sciences, information science, bioinformatics
Country of publisher	United States of America
Confidentiality degree	is not subject to a state or trade secret
Publication form	electronic version available online
WWW	URL
RIV identification code	RIV/00216224:14330/20:00115527
Organization unit	Faculty of Informatics
ISBN	978-1-61197-599-4
Doi	http://dx.doi.org/10.1137/1.9781611975994.136
UT WoS	000554408102017
Keywords in English	centered coloring; bounded expansion; planar graph; entropy compression
Tags	core_A, firank_1
Changed by	Changed by: RNDr. Pavel Šmerk, Ph.D., učo 3880. Changed: 14/6/2022 13:54.

Abstract
A vertex coloring c of a graph G is p-centered if for every connected subgraph H of G either c 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 >= 1, 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 O(p^3 log p) colors where the previous bound was O(p^19); (2) bounded degree graphs admit p-centered colorings with 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 non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require (p+t choose 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. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

Abstract

A vertex coloring c of a graph G is p-centered if for every connected subgraph H of G either c 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 >= 1, 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 O(p^3 log p) colors where the previous bound was O(p^19); (2) bounded degree graphs admit p-centered colorings with 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 non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require (p+t choose 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. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

Links
EF16_027/0008360, research and development project	Name: Postdoc@MUNI