Independent rainbow domination numbers of generalized Petersen graphs P(n,2) and P(n,3)

Boštjan Gabrovšek (Author), Aljoša Peperko (Author), Janez Žerovnik (Author)

Abstract

Dokažemo nove rezultate za neodvisna 2- in 3- mavrična dominacijska števila posplošenih Petersonovih grafov ▫$P(n,k)$▫ za nekatera števila ▫$n,k\in \mathbb{N}$▫. S primerno prilagoditvijo in uporabo dobro uveljavljene tehnike tropske algebre (algebre poti), dokažemo eksaktne vrednosti za 2-neodvisna mavrična dominacijska števila posplošenih Petersonovih grafov ▫$P(n,2)$▫ in ▫$P(n,3)$▫ in s tem dokažemo domnevo, ki so jo podali Shao in soavtorji. Nadalje izračunamo eksaktne vrednosti za neodvisna 2-mavrična dominacijska števila posplošenih Petersonovih grafov ▫$P(n,2)$▫. Metoda, ki jo uporabimo je razvita za mavrično dominacijo in posplošene Petersonove grafe. Z naravnimi modifikacijami pa lahko uporabljeno metodo uporabimo tudi za druge dominacijske invariante in za številne druge razredov grafov.

Keywords

neodvisna mavrična dominacija;neodvisno mavrično dominacijsko število;posplošeni Petersonovi grafi;tropska algebra;algebra poti;independent rainbow domination;independent rainbow domination number;generalized Petersen graphs;tropical algebra;path algebra;

Data

Language:	English
Year of publishing:	2020
Typology:	1.01 - Original Scientific Article
Organization:	UL FS - Faculty of Mechanical Engineering
UDC:	519.17
COBISS:	20459011
ISSN:	2227-7390
Views:	221
Downloads:	81
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Neodvisna mavrična dominacijska števila posplošenih Petersonovih grafov P(n,2) in P(n,3)
Secondary abstract:	We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs ▫$P(n,k)$▫ for certain values of ▫$n,k \in \mathbb{N}$▫. By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs ▫$P(n,2)$▫ and ▫$P(n,3)$▫ thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs ▫$P(n,2)$▫. The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori.
Secondary keywords:	neodvisna mavrična dominacija;neodvisno mavrično dominacijsko število;posplošeni Petersonovi grafi;tropska algebra;algebra poti;
Type (COBISS):	Article
Pages:	art. 996 (13 str.)
Volume:	ǂVol. ǂ8
Issue:	ǂiss. ǂ6
Chronology:	June 2020
DOI:	10.3390/math8060996
ID:	14075028