Fair reception and Vizing's conjecture

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).

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...

Vizing's conjecture: a survey and recent results

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 an...

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...