Majority colorings of sparse digraphs

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F21%3A00125298" target="_blank" >RIV/00216224:14330/21:00125298 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.37236/10067" target="_blank" >http://dx.doi.org/10.37236/10067</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.37236/10067" target="_blank" >10.37236/10067</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Majority colorings of sparse digraphs

Popis výsledku v původním jazyce

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree. Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most 6 or dichromatic number at most 3, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most 4. The benchmark case of r-regular digraphs remains open for r is an element of[5, 143]. Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring. We also give further evidence towards the existence of majority-3-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority (2 + epsilon)-coloring.

Název v anglickém jazyce

Majority colorings of sparse digraphs

Popis výsledku anglicky

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree. Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most 6 or dichromatic number at most 3, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most 4. The benchmark case of r-regular digraphs remains open for r is an element of[5, 143]. Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring. We also give further evidence towards the existence of majority-3-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority (2 + epsilon)-coloring.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

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

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název periodika

Electronic Journal of Combinatorics

ISSN

1077-8926

e-ISSN

—

Svazek periodika

28

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

17

Strana od-do

1-17

Kód UT WoS článku

000670355400001

EID výsledku v databázi Scopus

2-s2.0-85107211371