The edge span of T-coloring on graph Cnd

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 vertical bar f (x) - f (y)vertical bar is not an element of T whenever xy is an element of E (G). The edge span of a T-coloring is the maximum value of vertical bar f(x) - f(y)vertical bar over all edges xy, 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 dth 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 >= tk where n = m(d + 1) + r and r = ml + t with m >= 2, 0 <= r <= d, 0 <= l and 0 <= t <= m-1. (C) 2005 Elsevier Ltd. All rights reserved.