Almost all trees are almost graceful

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00524449" target="_blank" >RIV/67985840:_____/20:00524449 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1002/rsa.20906" target="_blank" >https://doi.org/10.1002/rsa.20906</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/rsa.20906" target="_blank" >10.1002/rsa.20906</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Almost all trees are almost graceful

Popis výsledku v původním jazyce

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1,2,…,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each γ>0 and for all n>n0(γ). Suppose that (i) the maximum degree of T is bounded by Oγ????(n∕log n), and (ii) the vertex labels are chosen from the set {1,2,…,⌈(1+γ)n⌉}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.

Název v anglickém jazyce

Almost all trees are almost graceful

Popis výsledku anglicky

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1,2,…,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each γ>0 and for all n>n0(γ). Suppose that (i) the maximum degree of T is bounded by Oγ????(n∕log n), and (ii) the vertex labels are chosen from the set {1,2,…,⌈(1+γ)n⌉}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Random Structures and Algorithms

ISSN

1042-9832

e-ISSN

—

Svazek periodika

56

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

40

Strana od-do

948-987

Kód UT WoS článku

000514232400001

EID výsledku v databázi Scopus

2-s2.0-85079856601