Improved bounds for centered colorings

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Improved bounds for centered colorings

Popis výsledku v původním jazyce

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 &gt;= 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.

Název v anglickém jazyce

Improved bounds for centered colorings

Popis výsledku anglicky

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 &gt;= 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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/EF16_027%2F0008360" target="_blank" >EF16_027/0008360: Postdoc@MUNI</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms

ISBN

9781611975994

ISSN

—

e-ISSN

—

Počet stran výsledku

15

Strana od-do

2212-2226

Název nakladatele

SIAM

Místo vydání

Not specified

Místo konání akce

Salt Lake City

Datum konání akce

1. 1. 2020

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000554408102017