The edge span of T-coloring on graph Cdn

Abstract Suppose G is a graph and T is a set of nonnegative integers that contains 0. A T -coloring of G is a nonnegative integer function f defined on V ( G ) such that | f ( x ) − f ( y ) | ∉ T whenever x y ∈ E ( G ) . The edge span of a T -coloring is the maximum value of | f ( x ) − f ( y ) | over all edges x y , and the T -edge span of G , esp T ( G ) , is the minimum edge span over all T -colorings of G . In this work, we continue to study the T -edge span of the d th power of the n -cycle C n , C n d , for T = { 0 , 1 , 2 , … , k − 1 } , prove that the condition gcd ( 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 esp T ( C n d ) for m ≥ t k where n = m ( d + 1 ) + r and r = m l + t with m ≥ 2 , 0 ≤ r ≤ d , 0 ≤ l and 0 ≤ t ≤ m − 1 .