An improvement on Vizing's conjecture

An Improved Inequality Related to Vizing's Conjecture

Vizing conjectured in 1963 that γ(G2H) > γ(G)γ(H) for any graphs G and H. A graph G is said to satisfy Vizing’s conjecture if the conjectured inequality holds for G and any graph H. Vizing’s conjecture has been proved for γ(G) 6 3, and it is known to hold for other classes of graphs. Clark and Suen in 2000 showed that γ(G2H) > 12γ(G)γ(H) for any graphs G and H. We give a slight improvement of t...

Vizing's conjecture: A two-thirds bound for claw-free graphs

Vizing's Conjecture for Graphs with Domination Number 3 - a New Proof

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 note we use a new, transparent approach to prove Vizing’s conjecture for graphs with domination number 3; that is, we prove that for any graph G with γ(G) = 3 and an arbitrary graph H, γ(G H) > 3γ(H).

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing’s conjecture on the domination number of Cartesian product graphs, by which we extend the well-known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharon...

On Vizing's conjecture

A dominating set D for a graph G is a subset of V (G) such that any vertex in V (G)−D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G2H Vizing’s conjecture [10] states that γ(G2H) ≥ γ(G)γ(H) for every pair of graphs G, H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prov...