T-COLORINGS, DIVISIBILITY AND THE CIRCULAR CHROMATIC NUMBER.

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