Some improved inequalities related to Vizing's conjecture

Let γ(G) be the domination number of a simple graph G and G□H be the Cartesian product of two simple graphs G and H. A function f:V(G)→{0,1,2} is a Roman dominating function (RDF) if for each vertex u∈V0,NG(u)∩V2≠∅, where Vi={u∈V(G):f(u)=i}. The Roman domination number γR(G) is the minimum weight f(V(G))=∑u∈V(G)f(u) among all RDFs of G. Vizing conjectured in 1963 that γ(G□H)≥γ(G)γ(H) for any graphs G and H. To this day, this conjecture remains open. In this paper, we show that for each pair of simple graphs G and H, γ(G□H)≥14γR(G)γR(H). This means that Vizing's conjecture holds for any pair of Roman graphs G and H. Moreover, we prove γR(G□H)≥γ(G)γ(H)+12min⁡{γ(G),γ(H)} if G or H is nonempty, which is a slight improvement of γR(G□H)≥γ(G)γ(H) obtained by Wu in 2013 [22].