T-colorings, divisibility and the circular chromatic number

LetTbe a $T-set$, i.e., a finite set of nonnegative integers satisfying $0∈T$,and$Gbe$ a graph. In the paper we study relations between theT-edge spansesp $T(G)$ and $espd⊙T(G$), wheredis a positive integer and $d⊙T={0≤t≤d(maxT+ 1) :d|t⇒t/d∈T}$.We show that $espd⊙T(G)$ =$despT(G)−r$, wherer, $0≤r≤d−1$, is aninteger that depends onTandG. Next we focus on the $caseT={0}$ and show that $espd⊙{0}(G) =⌈d(χc(G)−1)⌉$,where $χc(G)$ is the circular chromatic number ofG. This result allows us toformulate several interesting conclusions that include a new formula for thecircular chromatic $numberχc(G)$ = $1 + inf{espd⊙{0}(G)/d:d≥1}$ $2R$ and a proof that the formula for the $T-edge$ span of powers of cycles, statedas conjecture in [Y. Zhao, W. He and R. Cao,The edge span ofT-coloringon graphCdn, Appl. Math. Lett. 19 (2006) 647–651], is true