A note on circular colorings of edge-weighted digraphs

An edge-weighted digraph (G, ℓ) is a strict digraph G together with a function ℓ assigning a real weight ℓ_uv to each arc uv. (G, ℓ) is symmetric if uv is an arc implies that so is vu. A circular r-coloring of (G, ℓ) is a function φ assigning each vertex of G a point on a circle of perimeter r such that, for each arc uv of G, the length of the arc from φ(u) to φ(v) in the clockwise direction is at least ℓ_uv. The circular chromatic number χc(G, ℓ) of (G, ℓ) is the infimum of real numbers r such that (G, ℓ) has a circular r-coloring. Suppose that (G, ℓ) is an edge-weighted symmetric digraph with positive weights on the arcs. Let T be a {0, 1}-function on the arcs of G with the property that T(uv) + T(vu) = 1 for each arc uv in G. In this note we show that if, for each dicycle C of G satisfying, then (G, ℓ) has a circular r-coloring.