Vizing's conjecture: a survey and recent results

Boštjan Brešar (Author), Paul Dorbec (Author), Wayne Goddard (Author), Bert L. Hartnell (Author), Michael A. Henning (Author), Sandi Klavžar (Author), Douglas F. Rall (Author)

Abstract

Vizingova domneva iz leta 1968 trdi, da je dominacijsko število kartezičnega produkta dveh grafov vsaj tako veliko, kot je produkt dominacijskih števil faktorjev. V članku naredimo pregled različnih pristopov k tej osrednji domnevi iz teorije grafovske dominacije. Ob tem dokažemo tudi nekaj novih rezultatov. Tako so na primer pokazane nove lastnosti minimalnega protiprimera, dokazana je tudi nova spodnja meja za produkte grafov brez induciranega ▫$K_{1,3}$▫ s poljubnimi grafi. Skozi celoten članek so obravnavani pripadajoči odprti problemi, vprašanja in sorodne domneve.

Keywords

matematika;teorija grafov;kartezični produkt;dominacija;Vizingova domneva;mathematics;graph theory;Caretesian product;domination;Vizing's conjecture;

Data

Language:	English
Year of publishing:	2012
Typology:	1.02 - Review Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	519.17
COBISS:	16083801
ISSN:	0364-9024
Views:	426
Downloads:	20
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Unknown
Secondary title:	Vizingova domneva: pregled in novi rezultati
Secondary abstract:	Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper.
Secondary keywords:	matematika;teorija grafov;kartezični produkt;dominacija;Vizingova domneva;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 46-76
Volume:	Vol. 69
Issue:	iss. 1
Chronology:	2012
ID:	1475897