A NOTE ON CIRCULAR COLORINGS OF EDGE-WEIGHTED DIGRAPHS

Abstract

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 $uv$. $(\vec{G},\ell)$ 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 ${\sum_{uv\in E(\vec{C})} \ell_{uv}/\sum_{uv\in E(\vec{C})} T(uv)}\leq r$ for each dicycle $\vec{C}$ of $\vec{G}$ satisfying $0 \lt (\sum_{uv \in E(\vec{C})} \ell_{uv})\bmod r\lt \max \{\ell_{xy} + \ell_{yx}: xy \in E(\vec{G})\}$, then $(\vec{G},\ell)$ has a circular $r$-coloring. Our result generalizes the work of Zhu, J. Comb. Theory, Ser. B, 86 (2002), 109-113, and also strengthens the workof Mohar, J. Graph Theory, 43 (2003), 107-116.