Vizing's conjecture and the one-half argument

• Autorzy: Bert Hartnell , Douglas F. Rall
• Języki publikacji: EN

Abstrakty

EN

The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G☐H) ≥ γ(G)γ(H), where G☐H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G)-1 as well as the collection of graphs considered by Barcalkin and German in [1]. A crucial part of the proof depends on the well-known fact that the domination number of any connected graph of order at least two is no more than half its order.

Słowa kluczowe

EN

domination number; Cartesian product; Vizing's conjecture; clique

Kategorie tematyczne

• 05C70: Factorization, matching, partitioning, covering and packing
• 05C99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1995
• Tom: 15
• Numer: 2
• Strony: 205-216

Daty

wydano

1995

otrzymano

1995-05-25

Twórcy

autor

Bert Hartnell

• Saint Mary's University, Halifax, Nova Scotia, Canada B3H 3C3

autor

Douglas F. Rall

• Furman University, Greenville, SC 29613, U.S.A.

Bibliografia

• [1] A.M. Barcalkin and L.F. German, The external stability number of the Cartesian product of graphs, Bul. Akad. Stiince RSS Moldoven 1 (1979) 5-8.
• [2] M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs (Prindle, Weber & Schmidt International Series, 1979).
• [3] B.L. Hartnell and D.F. Rall, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96.
• [4] M.S. Jacobson and L.F. Kinch, On the domination of the products of graphs II: trees, J. Graph Theory 10 (1986) 97-106, doi: 10.1002/jgt.3190100112.
• [5] V.G. Vizing, The Cartesian product of graphs, Vyc. Sis. 9 (1963) 30-43.

Identyfikatory

DOI

10.7151/dmgt.1018