Edge-decomposing graphs into coprime forests

Kód výsledku v IS VaVaI

RIV/00216208:11320/21:10438301 - isvavai.cz

Výsledek na webu

https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=wBGGcbCYw0

DOI - Digital Object Identifier

10.1002/jgt.22638

Jazyk výsledku

angličtina

Název v původním jazyce

Edge-decomposing graphs into coprime forests

Popis výsledku v původním jazyce

The Barat-Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T, there is a constant c T such that every c T-edge-connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T-decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in Bensmail et al. (2019) that when T is a path with k edges, there is a constant d k such that every 24-edge-connected graph G with size divisible by k and minimum degree d k has a T-decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F-decomposition provided that the size of F divides the size of G (no connectivity is required). A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T-decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3.

Název v anglickém jazyce

Edge-decomposing graphs into coprime forests

Popis výsledku anglicky

The Barat-Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T, there is a constant c T such that every c T-edge-connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T-decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in Bensmail et al. (2019) that when T is a path with k edges, there is a constant d k such that every 24-edge-connected graph G with size divisible by k and minimum degree d k has a T-decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F-decomposition provided that the size of F divides the size of G (no connectivity is required). A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T-decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3.

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

GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)

Návaznosti

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

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název periodika

Journal of Graph Theory

ISSN

0364-9024

e-ISSN

—

Svazek periodika

97

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

13

Strana od-do

21-33

Kód UT WoS článku

000585798000001

EID výsledku v databázi Scopus

2-s2.0-85094641212