On defining numbers of circular complete graphs

AbstractLet d(σ) stand for the defining number of the colouring σ. In this paper we consider dmin=minγd(γ) and dmax=maxγd(γ) for the onto χ-colourings γ of the circular complete graph Kn,d. In this regard we obtain a lower bound for dmin(Kn,d) and we also prove that this parameter is asymptotically equal to χ-1. Also, we show that when χ⩾4 and s≠0 then dmax(Kχd-s,d)=χ+2s-3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G