T-COLORINGS, DIVISIBILITY AND THE CIRCULAR CHROMATIC NUMBER

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 is an element of T, and G be a graph. In the paper we study relations between the T-edge spans esp(T)(G) and esp(d circle dot T)(G), where d is a positive integer and d circle dot T = {0 <= t <= d (max T + 1) : d vertical bar t double right arrow t/d is an element of T}. We show that esp(d circle dot T)(G) = desp(T) (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 esp(d circle dot{0})(G) = inverted right perpendiculard(chi c(G) - 1)inverted left perpendicular, where chi(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 chi(c)(G) = 1 + inf {esp(d circle dot{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 C-n(d), Appl. Math. Lett. 19 (2006) 647-651], is true.