The edge span of TT-coloring on graph Cnd

Suppose GG is a graph and TT is a set of nonnegative integers that contains 0. A TT-coloring of GG is a nonnegative integer function ff defined on V(G)V(G) such that |f(x)−f(y)|∉T|f(x)−f(y)|∉T whenever xy∈E(G)xy∈E(G). The edge span of a TT-coloring is the maximum value of |f(x)−f(y)||f(x)−f(y)| over all edges xyxy, and the TT-edge span of GG, espT(G), is the minimum edge span over all TT-colorings of GG. In this work, we continue to study the TT-edge span of the ddth power of the nn-cycle CnCn, Cnd, for T={0,1,2,…,k−1}T={0,1,2,…,k−1}, prove that the condition gcd(n,d+1)=1gcd(n,d+1)=1 in the upper bound theorem provided by Hu, Juan and Chang is not necessary, give another lower bound, and find the exact value of espT(Cnd) for m≥tkm≥tk where n=m(d+1)+rn=m(d+1)+r and r=ml+tr=ml+t with m≥2m≥2, 0≤r≤d0≤r≤d, 0≤l0≤l and 0≤t≤m−10≤t≤m−1.