Common extremal graphs for three inequalities involving domination parameters

Let δ(G)ý, ýΔ(G) and γ(G)ý ýbe the minimum degreeý, ýmaximum degree andý ýdomination number of a graph G=(V(G)ý,ýE(G))ý, ýrespectivelyý. ýA partition of V(G)ý, ýall of whose classes are dominating sets in Gý, ýis called a domatic partition of Gý. ýThe maximum number of classes ofý ýa domatic partition of G is called the domatic number of Gý, ýdenoted d(G)ý. ýIt is well known thatý ýd(G)≤δ(G)ýý1ý, ýd(G)γ(G)≤|V(G)| \cite{ch}ý, ýand |V(G)|≤(Δ(G)ýý1)γ(G) \cite{berge}ý. ýIn this paperý, ýwe investigate the graphs G for whichý ýall the above inequalities become simultaneously equalitiesý.