\input zb-basic \input zb-ioport \iteman{io-port 05990178} \itemau{Lin, Wu-Hsiung; Yeh, Hong-Gwa} \itemti{A note on circular colorings of edge-weighted digraphs.} \itemso{Taiwanese J. Math. 15, No. 5, 2159-2167 (2011).} \itemab Summary: An edge-weighted digraph $(\vec{G},\ell)$ is a strict digraph $\vec{G}$ together with a function $\ell$ assigning a real weight $\ell_{uv}$ to each arc $uv \dot (\vec{G},\ell)$ is symmetric if $uv$ is an arc implies that so is $vu$. A circular $r$-coloring of $(\vec{G},\ell)$ is a function $\varphi$ assigning each vertex of $\vec{G}$ a point on a circle of perimeter $r$ such that, for each arc $uv$ of $\vec{G}$, the length of the arc from $\varphi(u)$ to $\varphi(v)$ in the clockwise direction is at least $\ell_{uv}$. The circular chromatic number $\chi_c(\vec{G},\ell)$ of $(\vec{G},\ell)$ is the infimum of real numbers $r$ such that $(\vec{G},\ell)$ has a circular $r$-coloring. Suppose that $(\vec{G},\ell)$ is an edge-weighted symmetric digraph with positive weights on the arcs. Let $T$ be a $\{0,1\}$-function on the arcs of $\vec{G}$ with the property that $T(uv)+T(vu)=1$ for each arc $uv$ in $\vec{G}$. In this note we show that if ${\displaystyle\sum_{uv\in E(\vec{C})} \ell_{uv}/\sum_{uv\in E(\vec{C})} T(uv)}\leq r$ for each dicycle $\vec{C}$ of $\vec{G}$ satisfying $\displaystyle 0< \left(\sum_{uv\in E(\vec{C})} \ell_{uv}\right) \mod r< \max\left\{\ell_{xy}+\ell_{yx}:xy\in E(\vec{G})\right\}$, then $(\vec{G},\ell)$ has a circular $r$-coloring. Our result generalizes the work of {\it X. Zhu} [J. Comb. Theory, Ser. B 86, No. 1, 109--113 (2002; Zbl 1025.05028)], and also strengthens the work of {\it B. Mohar} [J. Graph Theory 43, No. 2, 107--116 (2003; Zbl 1014.05026)]. \itemrv{~} \itemcc{} \itemut{circular chromatic number; edge-weighted digraph; } \itemli{http://tjm.math.ntu.edu.tw/index.php/TJM/article/view/252} \end