T-colorings, divisibility and circular chromatic number

Let T be a T-set, i.e., finite set of nonnegative integers satisfying 0 ? T, and G graph. In the paper we study relations between T-edge spans espT (G) and espd?T (G), where d is positive integer and d ? = {0 ? t (max + 1): |t ? t/d T} . We show that espd?T (G) espT (G) ? r, r 1, an integer depends on G. Next focus case {0} and show that espd?{0} (G) ?d(?c(G) 1)?, where ?c(G) circular chromatic number This result allows us to formulate several interesting conclusions include new formula for the circular number ?c(G) 1 inf espd?{0} (G)/d: ? 1 proof span powers cycles, stated as conjecture in [Y. Zhao, W. He R. Cao, The edge T-coloring on graph C d n , Appl. Math. Lett. 19 (2006) 647–651], true.