A Class of Graphs Approaching Vizing's Conjecture

Authors

Aziz Contractor, Clayton State UniversityFollow
Elliot Krop, Clayton State UniversityFollow

Publication Date

2016

Abstract

For any graph G=(V,E), a subset S ⊆ V dominates G if all vertices are contained in the closed neighborhood of S, that is N[S]=V. The minimum cardinality over all such S is called the domination number, written γ(G). In 1963, V.G. Vizing conjectured that γ(G □ H) ≥ γ(G)γ(H) where □ stands for the Cartesian product of graphs. In this note, we define classes of graphs Α_n, for n≥0, so that every graph belongs to some such class, and Α₀ corresponds to class A of Bartsalkin and German. We prove that for any graph G in class Α₁, γ(G□H)≥ γ(G)-√γ(G)γ(H).

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Contractor, Aziz and Krop, Elliot (2016) "A Class of Graphs Approaching Vizing's Conjecture," Theory and Applications of Graphs: Vol. 3: Iss. 1, Article 4.
DOI: 10.20429/tag.2016.030104
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol3/iss1/4