Abstract
Vizing's conjecture states that the domination number of the Cartesian product of graphs is at least the product of the domination numbers of the two factor graphs. In this note we improve the recent bound of Breŝar by applying a technique of Zerbib to show that for any graphs G and H, γ(G x H)≥ γ (G) 2/3(γ(H)-ρ(H)+1), where γ is the domination number, ρ is the 2-packing number, and x is the Cartesian product.
Highlights
Introduction and DefinitionsOne of the central problem in domination theory and product graphs is Vizing’s conjecture [5] which states that for any graphs G and H, (1.1)γ(G H) ≥ γ(G)γ(H).In this formulation, γ(G) is the domination number of G and G H is the Cartesian product of G and H.The truth of this statement is known for various classes of graphs
We show that for any two graphs G and H, (1.3)
Since we assumed |D ∩ Gh| < rh, such a set contains fewer than k vertices. This leads to the contradiction that there exists a subset of vertices of D of size less than k which can be projected onto G to form a dominating set
Summary
Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/tag Part of the Discrete Mathematics and Combinatorics Commons. Recommended Citation Wolff, Kimber (2020) "An improvement in the two-packing bound related to Vizing's conjecture," Theory and Applications of Graphs: Vol 7 : Iss. 1 , Article 5.
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