Improved bounds for centered colorings
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F20%3A00115527" target="_blank" >RIV/00216224:14330/20:00115527 - isvavai.cz</a>
Result on the web
<a href="https://epubs.siam.org/doi/abs/10.1137/1.9781611975994.136" target="_blank" >https://epubs.siam.org/doi/abs/10.1137/1.9781611975994.136</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/1.9781611975994.136" target="_blank" >10.1137/1.9781611975994.136</a>
Alternative languages
Result language
angličtina
Original language name
Improved bounds for centered colorings
Original language description
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.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
—
OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/EF16_027%2F0008360" target="_blank" >EF16_027/0008360: Postdoc@MUNI</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms
ISBN
9781611975994
ISSN
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e-ISSN
—
Number of pages
15
Pages from-to
2212-2226
Publisher name
SIAM
Place of publication
Not specified
Event location
Salt Lake City
Event date
Jan 1, 2020
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
000554408102017